Spatial Spread and Front Propagation Dynamics of Nonlocal Monostable
Equations in Periodic Habitats
by
Aijun Zhang
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
Aug 06, 2011
Keywords: Monostable equation; nonlocal dispersal; spreading speed; traveling wave
solution; principal eigenvalue; principal eigenfunction.
Copyright 2011 by Aijun Zhang
Approved by
Wenxian Shen, Chair, Professor of Mathematics and Statistics
Georg Hetzer, Professor of Mathematics and Statistics
Yanzhao Cao, Associate Professor of Mathematics and Statistics
Bertram Zinner, Associate Professor of Mathematics and Statistics
Abstract
This dissertation is concerned with spatial spread and front propagation dynamics of
monostable equations with nonlocal dispersal in spatially periodic habitats. Such equations
arise in modeling the population dynamics of many species which exhibit nonlocal internal
interactions and live in spatially periodic habitats. The main results of the dissertation
consist of the following four parts.
Firstly, we establish a general principal eigenvalue theory for spatially periodic nonlocal
dispersal operators. Some su cient conditions are provided for the existence of principal
eigenvalue and its associated positive eigenvector for such dispersal operators. It shows that a
spatially periodic nonlocal dispersal operator has a principal eigenvalue for the following three
special but important cases: (i) the nonlocal dispersal is nearly local; (ii) the periodic habitat
is nearly globally homogeneous or (iii) it is nearly homogeneous in a region where it is most
conducive to population growth. It also provides an example which shows that in general, a
spatially periodic nonlocal dispersal operator may not possess a principal eigenvalue, which
reveals some essential di erence between random dispersal and nonlocal dispersal. The
principal eigenvalue theory established in this dissertation provides an important tool for
the study of the dynamics of nonlocal monostable equations and is of also great importance
in its own.
Secondly, applying the principal eigenvalue theory for nonlocal dispersal operators and
comparison principle for sub- and super-solutions, we obtain one of the important features
for monostable equations, that is, the existence, uniqueness, and global stability of spatially
periodic positive stationary solutions to a general spatially periodic nonlocal monostable
equation. In spite of the use of the principal eigenvalue theory for nonlocal dispersal operators
in the proof, this feature is generic for nonlocal monostable equations in the sense it is
ii
independent of the existence of the principal eigenvalue of the linearized nonlocal dispersal
operator at the trivial solution of the monostable equation, which is of great biological
importance.
Thirdly, applying the principal eigenvalue theory for nonlocal dispersal operators and
comparison principle for sub- and super-solutions, we obtain another important feature for
monostable equations, that is, the existence of a spatial spreading speed of a general spatially
periodic nonlocal equation in any given direction, which characterizes the speed at which a
species invades into the region where there is no population initially in the given direction.
It is also seen that this feature is generic for nonlocal monostable equations in the same sense
as above. Moreover, it is shown that spatial variation of the habitat speeds up the spatial
spread of the population.
Finally, this dissertation also deals with front propagation feature for monostable equa-
tions with non-local dispersal in spatially periodic habitats. It is shown that a spatially
periodic nonlocal monostable equation has in any given direction a unique stable spatially
periodic traveling wave solution connecting its unique positive stationary solution and the
trivial solution with all propagating speeds greater than the spreading speed in that direc-
tion for the special but important cases mentioned above, that is, (i) the nonlocal dispersal
is nearly local; (ii) the periodic habitat is nearly globally homogeneous or (iii) it is nearly
homogeneous in a region where it is most conducive to population growth. It remains open
whether this feature is generic or not for spatially periodic nonlocal monostable equations.
iii
Acknowledgments
It would have been next to impossible to nish my dissertation without the guidance of
my committee members and support from my family and wife. I would like to gratefully and
sincerely thank my adviser, Dr. Wenxian Shen, for her excellent guidance for doing research,
patience, and caring. I would also like to thank my committee members, Dr. Yanzhao Cao,
Dr. Georg Hetzer, and Dr. Bertram Zinner for guiding my research for the past several
years. I would also like to thank my parents, my younger sister, and younger brother for
their support and encouragement. Finally, I would like to thank my wonderful wife, Le Wu
for her continued support and understanding. She was always there cheering me up and
stood with me through the good times and the bad. I would like to thank the NSF for the
nancial support (NSF-DMS-0907752).
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Notations, Assumptions, De nitions and Main Results . . . . . . . . . . . . . . 9
2.1 Notations, Assumptions and De nitions . . . . . . . . . . . . . . . . . . . . . 9
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Comparison Principle and Sub- and Super-solutions . . . . . . . . . . . . . . . . 23
3.1 Solutions of Evolution Equation and Semigroup Theory . . . . . . . . . . . . 23
3.2 Sub- and Super-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Comparison Principle and Monotonicity . . . . . . . . . . . . . . . . . . . . 25
3.4 Convergence on Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Spectral Theory of Dispersal Operators . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Evolution Operators and Eigenvalue Problems . . . . . . . . . . . . . . . . . 33
4.2 Existence of the Principal Eigenvalue . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Spectral Radius of the Truncated Evolution Operator . . . . . . . . . . . . . 46
4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 E ects of Spatial Variation on Principal Eigenvalue . . . . . . . . . . . . . . 55
5 Positive Stationary Solutions of Spatially Periodic Nonlocal Monostable Equations 58
6 Spreading Speeds of Spatially Periodic Nonlocal Monostable Equations . . . . . 62
6.1 Spreading Speed Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 Spreading Speeds under the Assumption of the Existence of a Principal Eigen-
value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
v
6.3 Spreading Speeds in the General Case . . . . . . . . . . . . . . . . . . . . . . 89
6.4 E ects of Spatial Variations on Spreading Speeds . . . . . . . . . . . . . . . 92
7 Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations . 94
7.1 Sub- and Super-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Existence and Uniqueness of Traveling Wave Solutions . . . . . . . . . . . . 101
7.3 Stability of Traveling Wave Solutions . . . . . . . . . . . . . . . . . . . . . . 116
8 Concluding Remarks, Open Problems, and Future Plan . . . . . . . . . . . . . . 123
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
vi
Chapter 1
Introduction
This dissertation is devoted to the study of spatial spread and front propagation dynam-
ics of monostable equations with nonlocal dispersal in spatially periodic habitats. Monostable
equations are widely used to model the population dynamics of many species in biology and
ecology. In general, such an equation is of the following form,
@u
@t = Du+u(t;x)f(x;u(t;x)); x2 R
N; (1.1)
where u(t;x) represents the population density of a species at time t and spatial location
x, D is a dispersal operator which measures the di usion or redistribution of the species,
> 0 is the dispersal rate, and the term f measures the growth rate of the population of the
species and satis es the so called monostablility assumptions (that is, f(x;u) < 0 for u 1,
@f
@u(x;u) < 0 for u 0 and u 0 is linearly unstable in proper sense), the domain R
N
(or ZN) may be bounded or unbounded. Without loss of generality, can be chosen 1 by
rescaling time t and changing f.
Among the dispersal operators often adopted in literature are nonlocal, random or
local, and discrete dispersal operators. In particular, (1.1) with D being a nonlocal dispersal
operator, that is,
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +u(t;x)f(x;u(t;x)); x2 ; (1.2)
is widely used to model the population dynamics of a species in which the movements or
interactions of the organisms occur between non-adjacent spatial locations, where k : RN !
R+ is a nonlocal dispersal kernel function with RRN k(x)dx = 1. Classically, one assumes
that the internal interaction of the organisms in a species is random and local (i.e. species
1
moves randomly between the adjacent spatial locations), which leads to (1.1) with D = ,
that is, reaction-di usion equations of the following form,
@u
@t = u+uf(x;u); x2 : (1.3)
Another dispersal strategy is nearest neighbor interaction in a patchy environment modeled
by the lattice ZN. This leads to (1.1) with D being a discrete dispersal operator, that is,
the following lattice system of ordinary di erential equations
_uj =
X
k=(k1;k2; ;kN)2K
(uj+k uj) +ujf(j;uj); j2 (1.4)
where K =f(k1;k2; ;kN)2ZNjk21 + +k2N = 1g.
Nonlocal, random, and discrete dispersal evolution equations are then of great interests
in their own. They are also related to each other. For example, (1.4) can be viewed as a
spatial discretization of (1.3). In order to indicate some relationship between nonlocal and
random dispersal, we take k(z) = 1 N ~k(z= ) for some > 0 and ~k( ) : RN !R+ which is
smooth, symmetric, supported on B(0;1) :=fx2RNjkxk< 1g, and RRN ~k(x)dx = 1. Then
for any smooth function u(x),
Z
RN
k(y x)u(y)dy u(x)
=
Z
RN
~k(z)
"
u(x) + (ru(x) z) +
2
2
NX
i;j=1
uxixj(x)zizj +O( 3)
#
dz u(x)
=
2
2N
Z
RN
~k(z)kzk2dz u(x) +O( 3):
Hence, the dispersal operator u 7! RRN k( y)u(y)dy u( ) \behaves" the same as the
operator u7! 22N RRN ~k(z)kzk2dz u for 1, and plays basically the role of a dispersal
rate.
2
In this dissertation, we will focus on spatially periodic nonlocal monostable equations in
unbounded domains, that is, equations of the form (1.2) with f(x;u) being periodic in x and
being of the monostable properties (see (H2) and (H3) in Charter 2 for detail). We remark
that heterogeneities are present in many biological and ecological models. The periodicity
of f(x;u) in x takes into account the periodic heterogeneities of the media of the underlying
systems and monostablility assumptions re ect the natural feature for population growth
models.
Common and central dynamical issues about dispersal monostable equations in un-
bounded domains include the understanding of spatial spread and front propagation dynam-
ics. Here are two most fundamental dynamical problems associated to the spatial spread
and front propagation dynamics of monostable equations: how fast the population spreads
as time evolves? are there solutions which preserve the shape and propagate at some speed
along certain direction?
The study of spatial spread and front propagation dynamics of monostable equations
traces back to Fisher [19] and Kolmogorov, Petrowsky, and Piscunov [39]. In the pioneering
works of Fisher [19] and Kolmogorov, Petrowsky, Piscunov [39], they studied the spatial
spread and front propagation dynamics of the following special case of (1.3)
@u
@t =
@2u
@x2 +u(1 u); x2R: (1.5)
Here u is the frequency of one of two forms of a gene. Fisher in [19] found traveling wave
solutions u(t;x) = (x ct), ( ( 1) = 1; (1) = 0) of all speeds c 2 and showed that
there are no such traveling wave solutions of slower speed. He conjectured that the take-
over occurs at the asymptotic speed 2. This conjecture was proved in [39] by Kolmogorov,
Petrowsky, and Piscunov, that is, they proved that for any nonnegative solution u(t;x) of
(1.5), if at time t = 0, u is 1 near 1 and 0 near 1, then limt!1u(t;ct) is 0 if c> 2 and 1 if
c< 2 (i.e. the population invades into the region with no initial population with speed 2).
3
Since then, the spatial speed and front propagation dynamics of (1.3) has been widely
studied (see [1], [2], [4], [5], [6], [18], [21], [26], [31], [36], [41], [42], [44], [47], [48], [51], [52],
[53], [61], [62], [63], and references therein). The spatial spreading dynamics of (1.5) has
been well extended to (1.3). To be more precise, assume that f(x;u) is periodic in x, that is
f(x + piei;u) = f(x;u) for some pi > 0 (i = 1;2; ;N), ei denotes the vector with a 1 in
the ith coordinate and 0?s elsewhere, and satis es the following monostablility assumptions:
f2C1(RN [0;1);R), sup
x2RN;u 0
@f(x;u)
@u < 0, f(x;u) < 0 for x2R
N and u 1, and the
principal eigenvalue of 8
>><
>>:
u+a0(x)u = u; x2RN
u(x+piei) = u(x); x2RN
is positive, where a0(x) = f(x;0). Without loss of generality, assume = 1. It has been
shown that (1.3) has exactly two spatially periodic equilibrium solutions, u = 0 and u = u+,
and u = 0 is linearly unstable and u = u+ is globally asymptotically stable with respect
to spatially periodic perturbations (which gives a reason which the above assumptions are
referred to monostability assumptions). Let 2SN 1 := f 2RNjk k = 1g. It has also
been shown that for every 2SN 1, there is a c ( )2R such that for every c c ( ), there
is a traveling wave solution connecting u+ and u 0 and propagating in the direction of
with speed c, and there is no such traveling wave solution of slower speed in the direction of
. The minimal wave speed c ( ) is of some important spreading properties, that is,
lim inft!1 inf
x ct
(u(t;x;u0) u+(x)) = 0 8cc ( );
for all nonnegative uniformly continuous bounded function u0 satisfying that u0(x) 0 for
some 0 > 0 and x2RN with x 1 and u0(x) = 0 for x2RN with x 1. Here
4
u(t;x;u0) denotes the solution of (1.3) with u(0;x;u0) = u0(x) and inf
x ct
( sup
x ct
) denotes the
in mum (supremum) taken over all the x2 RN satisfying that x ct (x ct) for
given 2SN 1 and c;t2R. Hence c ( ) is also called the spreading speed of (1.3) in the
direction of . Moreover, it has the following variational characterization. Let ( ; ) be the
eigenvalue of 8
>><
>>:
u 2 PNi=1 i @u@xi + (a0(x) + 2)u = u; x2RN
u(x+piei) = u(x); x2RN
(1.6)
with largest real part, where a0(x) = f(x;0) (it is well known that ( ; ) is real and
algebraically simple. ( ; ) is called the principal eigenvalue of (1.6) in literature). Then
c ( ) = inf >0 ( ; ) : (1.7)
(See [4], [5], [6], [41], [47], [48], [63] and references therein for the above mentioned properties).
Spatial spread and front propagation dynamics is also quite well studied for monostbale
equations with discrete dispersal. We refer to [9], [10], [24], [32], [54], [59], [60], [62], [63],
[64], [65], etc. for the study of spatial spread and front propagation dynamics dynamics of
monostable equations with discrete dispersal of the form (1.4).
The objective of this dissertation is to investigate the spatial spread and front propaga-
tion of spatially periodic nonlocal monostable equations of the form (1.2).
Recently, various dynamical problems related to the spatial spread and front propagation
dynamics of nonlocal dispersal equations of the form (1.2) have also been studied by many
authors. See, for example, [3], [8], [11], [13], [15], [22], [23], [28], [29], [33], [34], [35], [37],
[55], for the study of spectral theory for nonlocal dispersal operators and the existence,
uniqueness, and stability of nontrivial positive stationary solutions. See, for example, [12],
[14], [16], [40], [45], [49], [62], [63], for the study of entire solutions and the existence of
spreading speeds and traveling wave solutions connecting the trivial solution u = 0 and a
nontrivial positive stationary solution for some special cases of (1.2). In particular, if f(x;u)
5
is independent of x, then it is proved that (1.2) has a spreading speed c ( ) in every direction
of 2SN 1 (c ( ) is indeed independent of 2SN 1 in this case) and for every c c ( ),
(1.2) has a traveling wave solution connecting u+ and 0 and propagating in the direction of
with propagating speed c (see [12]).
However, most existing works on spatial spreading dynamics of monostable equations
with nonlocal dispersal deal with spatially homogeneous equations (i.e. f(x;u) in (1.2) is
independent of x). There is little understanding of the spatial spread and front propagation
dynamics of general nonlocal dispersal monostable equations. One major di erence between
(1.3) and (1.2) is that the solution operator of (1.3) in proper phase space is compact with
respect to the uniform convergence on bounded subsets of RN (i.e., is compact with respect
to open compact topology), whereas the solution operator of (1.2) in a usual phase space
does not exhibit such compactness features. It appears to be di cult to adopt many existing
methods for the study of spatial spread and front propagation dynamics of random dispersal
monostable equations in dealing with (1.2) in general due to the lack of compactness of the
solution operator and the spatial inhomogeneity of the nonlinearity. In fact, there is even
a lack of general principal eigenvalue theory for nonlocal dispersal operators and a lack of
positive stationary solutions of spatially periodic nonlocal monostable equations, which are
important tools/ingredients in the study of spatial spread and front propagation dynamics
of monostable equations.
In this dissertation, we will then rst establish a general principal eigenvalue theory
for spatially periodic nonlocal dispersal operators (see chapter 4). We show that a spatially
periodic nonlocal dispersal operator has a principal eigenvalue for following three special but
important cases: (i) the nonlocal dispersal is nearly local; (ii) the periodic habitat is nearly
globally homogeneous or (iii) it is nearly homogeneous in a region where it is most conducive
to population growth. It also provides an example which shows that in general, a spatially
periodic nonlocal dispersal operator may not possess a principal eigenvalue, which reveals
some essential di erence between random dispersal and nonlocal dispersal. The principal
6
eigenvalue theory established in this dissertation provides an important tool for the study of
the dynamics of nonlocal monostable equations and is of also great importance in its own.
Next, applying the principal eigenvalue theory for nonlocal dispersal operators and
comparison principle for sub- and super-solutions, we obtain one of the important features
for monostable equations, that is, the existence, uniqueness, and global stability of spatially
periodic positive stationary solutions to a general spatially periodic nonlocal monostable
equation (see chapter 5). In spite of the use of the principal eigenvalue theory for nonlocal
dispersal operators in the proof, this feature is generic for nonlocal monostable equations
in the sense it is independent of the existence of the principal eigenvalue of the linearized
nonlocal dispersal operator at the trivial solution of the monostable equation, which is of
great biological importance.
Furthermore, we then investigate the spatial spreading speeds of spatially periodic non-
local monostable equations (see chapter 6). Applying the principal eigenvalue theory for
nonlocal dispersal operators and comparison principle for sub- and super-solutions, we ob-
tain another important feature for monostable equations, that is, the existence of a spatial
spreading speed of a general spatially periodic nonlocal equation in any given direction,
which characterizes the speed at which a species invades into the region where there is no
population initially in the given direction. It is also seen that this feature is generic for non-
local monostable equations in the same sense as above. Moreover, it is shown that spatial
variation of the habitat speeds up the spatial spread of the population.
Finally, we deal with traveling wave solutions of monostable equations with non-local
dispersal in spatially periodic habitats (see chapter 7). It is shown that a spatially periodic
nonlocal monostable equation has in any given direction a unique stable spatially periodic
traveling wave solution connecting its unique positive stationary solution and the trivial
solution with all propagating speeds greater than the spreading speed in that direction for
the special but important cases mentioned above, that is, (i) the nonlocal dispersal is nearly
local; (ii) the periodic habitat is nearly globally homogeneous or (iii) it is nearly homogeneous
7
in a region where it is most conducive to population growth. It remains open whether this
feature is generic or not for spatially periodic nonlocal monostable equations.
The rest of the dissertation is organized as follows. In chapter 2, we state some standing
notations, assumptions, de nitions and the main results. In chapter 3, we develop some
basic tools or fundamental theory for the use in later chapters, such as semigroup theory,
comparison principle, sub- and super-solutions. We will investigate the spectral theory of
nonlocal dispersal operators in chapter 4. In chapter 5, we study the existence, uniqueness
and stability of stationary solutions of (1.2). In chapter 6, spatial spreading speeds of (1.2)
are investigated. In chapter 7, we study the existence,uniqueness and stability of the traveling
wave solutions of spatially periodic nonlocal monostable equations. The dissertation will end
up with remarks, open problems, and future plan in chapter 8.
8
Chapter 2
Notations, Assumptions, De nitions and Main Results
In this chapter, we introduce rst the standing notations, assumptions, and the de ni-
tions of principal eigenvalue of nonlocal dispersal operators, spatial spreading speeds, and
traveling wave solutions of spatially periodic nonlocal monostable equations. We then state
the main results of the dissertation.
2.1 Notations, Assumptions and De nitions
In this section, we introduce the standing notations, assumptions, and the de nitions of
principal eigenvalue of nonlocal dispersal operators, spatial spreading speeds, and traveling
wave solutions of spatially periodic nonlocal monostable equations.
Consider (1.2) with = RN, that is,
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +u(t;x)f(x;u(t;x)); x2RN:
We assume that f(x;u) is periodic in x, that is, there are pi > 0 (i = 1;2; ;N) such
that f(x + piei) = f(x;u) for all x2RN and u2R. We assume that the nonlocal kernel
function k( ) satis es the following assumption.
(H1) k( )2C1(RN;R+), RRN k(z)dz = 1, k(0) > 0 and RRN k(z)e kzkdz <1 for any > 0.
We remark that (H1) implies that the kernel function k( ) is actually a smooth probabil-
ity density function of some random variable X, and k( ) is strictly positive at the origin and
the expected value of e jXj is nite,that is, E(e jXj) <1. There are a lot of such examples.
9
For instance, the probability density functions of normal distributions and all smooth prob-
ability density functions which are positive at the origin and supported on a bounded set
satisfy (H1). The following is one example which has a bounded support: k(z) = 1 N
0
~k(z= 0)
~k(z) =
8>
><
>>:
C exp 1kzk2 1 for kzk< 1
0 for kzk 1;
(2.1)
where C > 0 is chosen such that RRN ~k(z)dz = 1.
Let
X =fu2C(RN;R)ju is uniformly continuous on RN and sup
x2RN
ju(x)j<1g (2.2)
with norm kukX = sup
x2RN
ju(x)j, and
X+ =fu2Xju(x) 0 8x2RNg: (2.3)
Let
Xp =fu2Xju(x+piei) = u(x) 8x2RN; i = 1;2; ;Ng (2.4)
and
X+p =fu2Xpju(x) 0 8x2RNg: (2.5)
Let I be the identity map on Xp, and K, a0( )I : Xp!Xp be de ned by
Ku (x) = Z
RN
k(y x)u(y)dy; (2.6)
(a0( )Iu)(x) = a0(x)u(x); (2.7)
where a0(x) = f(x;0).
10
Throughout this dissertation, a function h : RN R!R is said to be smooth if h(x;u)
is CN in x2RN and C1 in u2R. We assume that f satis es the following \monostablility"
assumptions:
(H2) f2C1(RN [0;1);R), sup
x2RN;u 0
@f(x;u)
@u < 0 and f(x;u) < 0 for x2R
N and u 1.
(H3) u 0 is linearly unstable in Xp, that is, 0 := supfRe j 2 (K I + a0( )I)g is
positive, where (K I +a0( )I) is the spectrum of the operator K I +a0( )I on Xp.
A typical example for f(x;u) is f(x;u) = a(x) u with a( ) 2X+p nf0g. Note that
(H2) and (H3) re ect the natural feature of population growth models.
Among the main techniques employed in this dissertation are the comparison principle,
sub- and super-solutions, and the principal eigenvalue theory of the eigenvalue problem,
K
; I +a( )I
v = v; v2X
p; (2.8)
where 2SN 1, 2R, and a( )2Xp, the operator a( )I has the same meaning as in (2.7)
with a0( ) being replaced by a( ), and K ; : Xp!Xp is de ned by
(K ; v)(x) =
Z
RN
e (y x) k(y x)v(y)dy: (2.9)
We point out the following relation between (1.2) and (2.8): if u(t;x) = e (x t) (x) with
2Xpnf0g is a solution of the linearization of (1.2) at u = 0,
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +a0(x)u(t;x); x2RN; (2.10)
where a0(x) = f(x;0), then is an eigenvalue of (2.8) with a( ) = a0( ) or K ; I +a0( )I
and v = (x) is a corresponding eigenfunction.
De nition 2.1 (Principal eigenvalue). Let (K ; I +a( )I) be the spectrum of K ; I +
a( )I on Xp.
11
(1) 0( ; ;a) := supfRe j 2 (K ; I +a( )I)g is called the principal spectrum point
of K ; I +a( )I.
(2) A number ( ; ;a)2R is called the principal eigenvalue of (2.8) or K ; I +a( )I
if it is an algebraically simple eigenvalue of K ; I + a( )I with an eigenfunction
v2X+p , and for every 2 (K ; I +a( )I)nf ( ; ;a)g, Re < ( ; ;a).
Observe that if the principal eigenvalue ( ; ;a) ofK ; I+a( )I exists, then ( ; ;a) =
0( ; ;a). If = 0, (2.8) is independent of and hence we put
0(a) := 0( ;0;a) 8 2SN 1: (2.11)
Due to the lack of compactness of the semigroup generated by K ; I + a( )I on Xp
and the inhomogeneity of a( ), the existence of a principal eigenvalue and eigenfunction of
(2.8) cannot be obtained from standard theory (e.g. the Krein-Rutman theorem). It should
be pointed out that recently the principal eigenvalue problem for nonlocal dispersal has been
studied by several authors (see [35], [37], [55], etc.). However, the existing results cannot
be applied directly to (2.8). We will hence develop a principal eigenvalue theory for (2.8)
or K ; I + a( )I in chapter 4. At some places, we make the following assumption on the
existence of principal eigenvalues.
(H4) ( ; ;a) exists for all 2SN 1 and 0.
In the following, inf
x r
( sup
x r
) represents the in mum (supremum) taken over all the
x2RN satisfying that x r for given 2SN 1 and r 2R. Similarly, the notations
inf
x ct
, inf
jx j ct
, inf
kxk ct
( sup
x ct
, sup
jx j ct
, sup
kxk ct
) represent the in ma (suprema) taken over all the
x2RN satisfying the inequalities in the notations for given 2SN 1 and c;t2R. For a
12
given 2SN 1, and
X+( ) =fu2X+j lim infr! 1 inf
x r
u(x) > 0;
u(x) = 0 for x2RN with x 1g: (2.12)
It follows from the general semigroup approach (see [27] or [50]) that (1.2) has a unique
(local) solution u(t;x;u0) with u(0;x;u0) = u0(x) for every u0 2X. Moreover, a comparison
principle in the usual sense holds for solutions of (1.2), and u(t;x;u0) exists for all t 0 if
u0 2X+ (see Proposition 3.1).
De nition 2.2 (Spatial spreading speed). Assume that (H1) - (H3) are ful lled and that
2SN 1. We call a number c ( )2R the spatial spreading speed of (1.2) in the direction
of if the following properties are satis ed:
lim inft!1 inf
x ct
u(t;x;u0) > 0 8cc ( )
for every u0 2X+( ).
Observe that our de nition coincides with the notion ofc ( ) in [62] provided thatf(x;u)
is independent of x. The construction based de nition used in [44], [62], [63] is di erent in
the sense that our de nition does not guarantee the existence of c ( ). In fact, we focus in
this dissertation on investigating the existence and characterization of c ( ) for 2SN 1.
To this end, let
~X =fu : RN !Rju is Lebesgue measurable and boundedg (2.13)
endowed with the norm kuk~X = sup
x2RN
ju(x)j and
13
~X+ =fu2 ~Xju(x) 0 8x2RNg: (2.14)
Observe that Xp X ~X.
To study the spatial spreading and front propagation dynamics of (1.2), we sometime
need to consider the space shifted equation of (1.2)
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +u(t;x)f(x+z;u(t;x)); x2RN (2.15)
where z 2 RN. Let u(t;x;u0;z) be the solution of (2.15) with u(0;x;u0;z) = u0(x) for
u0 2X.
By general semigroup theory (see [27] and [50]), for any u0 2 ~X and z2R, (2.15) has
a unique (local) solution u(t; ) 2 ~X with u(0;x) = u0(x). Let u(t;x;u0;z) be the solution
of (2.15) with u(0;x;u0;z) = u0(x). Note that if u0 2Xp (resp. X), then u(t; ;u0;z)2Xp
(resp. X). If u0 2 ~X+, then u(t;x;u0;z) exists for all t 0.
De nition 2.3 (Entire solution). A measurable function u : R RN :!R is call an entire
solution of (1.2) if u(t;x) is di erentiable in t 2 R and satis es (1.2) for all t 2 R and
x2RN.
De nition 2.4 (Traveling wave solution). Suppose that (1.2) has a spatially periodic positive
stationary solution u = u+( )2X+P nf0g.
(1) An entire solution u(t;x) of (1.2) is called a traveling wave solution connecting u+( )
and 0 and propagating in the direction of with speed c if there is a bounded measurable
function : RN RN !R+ such that u(t; ; ( ;z);z) exists for all t2R,
u(t;x) = u(t;x; ( ;0);0) = (x ct ;ct ) 8t2R; x2RN; (2.16)
u(t;x; ( ;z);z) = (x ct ;z +ct ) 8t2R; x;z2RN; (2.17)
lim
x ! 1
(x;z) u+(x+z) = 0; lim
x !1
(x;z) = 0 uniformly in z2RN; (2.18)
14
(x;z x) = (x0;z x0) 8x;x02RN with x = x0 ; (2.19)
and
(x;z +piei) = (x;z) 8x;z2RN: (2.20)
(2) A bounded measurable function : RN RN !R+ is said to generate a traveling wave
solution of (1.2) in the direction of with speed c if it satis es (2.17)-(2.20).
Remark 2.1. Suppose that u(t;x) = (x ct ;ct ) is a traveling wave solution of (1.2)
connecting u+( ) and 0 and propagating in the direction of with speed c. Then u(t;x) can
be written as
u(t;x) = (x ct;x) (2.21)
for some : R RN !R satisfying that ( ;z + piei) = ( ;z), lim ! 1 ( ;z) = u+(z),
and lim !1 ( ;z) = 0 uniformly in z 2RN. In fact, let ( ;z) = (x;z x) for x2RN
with x = . Observe that ( ;z) is well de ned and has the above mentioned properties.
In some literature, the form (2.21) is adopted for spatially periodic traveling wave solutions
(see [41], [46], [63], and references therein).
2.2 Main Results
In this section, we state the main results of the dissertation. The rst two theorems are
about the principal eigenvalue of (2.8).
Theorem A. (Su cient conditions for the existence of principal eigenvalues)
(1) Support that k(z) = 1 N ~k(z ) for some > 0 and ~k( ) with supp(~k) = B(0;1) := fz2
RNjkzk< 1g. There is 0 > 0 such that for every 0 < 0, the principal eigenvalue
( ; ;a) of (2.8) exists for all 2SN 1 and 2R.
(2) If a(x) satis es that max
x2RN
a(x) min
x2RN
a(x) < with = minfRz <0k(z)dzj 2SN 1g,
then the conclusions in (1) hold.
15
(3) If a2XP\CN(RN) and the partial derivatives of a(x) up to order N 1 at some x0
are zero, where x0 is such that a(x0) = max
x2RN
a(x), then the conclusions in (1) hold.
Let ( ; ;a) be the principal eigenvalue of K ; I +a( ). Note that if = 0, (2.8) is
independent of and hence we put
(a) := ( ;0;a) 8 2SN 1 (2.22)
(if ( ;0;a) exists).
Let a = 1jDjRDf(x;0)dx with D =
NY
i=1
[0;pi] and jDj =
NY
i=1
pi where the period vector
p = (p1;:::;pN).
Theorem B. (In uence of spatial variation) Assume that ( ; ;a(x)) of K ; I + a( )I
exists for any 2SN 1 and 2R. Then ( ; ;a(x)) ( ; ; a) for any 2SN 1 and
2R. ( ; ;a(x)) = ( ; ; a) for some 2SN 1 and 2R i a(x) a.
We remark that the proof of Theorem A , the existence part of the principal eigenvalue,
relies on techniques from the perturbation theory of Burger [7] (see [7, Proposition 2.1 and
Theorem 2.2]) and on the arguments in [37, Theorem 2.6]. However, special care is required
in view of the dependence of K ; on 2SN 1 and 2R. Note that the conclusions are
independent of 2SN 1 and 2R (i.e. for proper > 0 and a, ( ; ;a) exists for every
2 SN 1 and 2 R). Theorem A (1) is proved in [37] for = 0 with the assumption
that k( ) is symmetric with a bounded support, that is, k(z) = k( z) supported on a ball
for z2RN. We extended A(1) to general kernel for all 2R. Theorem A(3) will play an
important role in proving the existence of positive stationary solution and generic spreading
speeds.
As it is well known, the principal eigenvalue of a random or local dispersal operator
always exists. By Theorem A(1), if the nonlocal dispersal operatorK ; I +a( )I is nearly
local in the sense that the dispersal distance is su ciently small, then we obtain a similar
principal eigenvalue theory as for random dispersal operators.
16
Observe that K ; : Xp ! Xp is a compact and positive operator. If a(x) a is
independent of x, then it is not di cult to see that ( ; ;a) := a 1 +RRN e z k(z)dz is
the principal eigenvalue ofK ; I+a( )I for 2SN 1, and 2R. By Theorem A(2)(3), if
a( ) has certain homogeneity features, then the nonlocal dispersal operator K ; I +a( )I
also possesses a principal eigenvalue. More precisely, Theorem A (2) shows that if a(x)
is nearly globally homogeneous or globally at in the sense that max
x2RN
a(x) min
x2RN
a(x) < ,
then the principal eigenvalue ( ; ;a) of the nonlocal dispersal operator K ; I + a( )I
exists for 2 SN 1, and 2 R. Note that if K ; I + a( )I in (2.8) is replaced by
[K ; I]+a( )I with a general positive dispersal rate > 0, Theorem A (2) holds provided
that max
x2RN
a(x) min
x2RN
a(x) < . If k( ) is symmetric, then can be chosen 1. So A (2) holds
provided that max
x2RN
a(x) min
x2RN
a(x) < , which means biologically that the variation in the
habitat is less than the dispersal rate of the nonlocal dispersal operator K ; I. We say
a( ) is nearly homogeneous or at in some region where it is most conducive to population
growth in the zero-limit population (which will be referred to as nearly locally homogeneous
in the following) if all partial derivatives of a(x) up to order N 1 are zero at some x0 with
a(x0) = max
x2RN
a(x). Theorem A (3) shows that if a( ) is nearly locally homogenous, then for
any 2SN 1, and 2R, the principal eigenvalue ( ; ;a) of K ; I + a( )I exists, too.
It should be pointed out that a(x) is nearly globally homogeneous may not imply that it is
nearly locally homogeneous.
Clearly, the \ atness" condition for a(x) in Theorem A (3) is always satis ed for N = 1
or 2. Hence when N = 1 or 2, the principal eigenvalue of K ; I + a( )I exists for all
2 SN 1, 2 R. In general, if N 3 , the principal eigenvalue of (2.8) may not exist
(see example in chapter 4). This reveals an essential di erence between nonlocal dispersal
operators and random dispersal operators.
How do spatial variations a ect the principal eigenvalue (if exists)? In biological sense,
Theorem B shows that spatial variation cannot reduce the principal eigenvalue of a dispersal
17
operator with nonlocal dispersal and periodic boundary condition, and indeed it is increased
except for degenerate cases.
After we established the spectral theory of the nonlocal dispersal operators, we can em-
ploy the comparison principle and construct sub- super-solutions to investigate the existence,
uniqueness and stability of positive equilibrium solutions of (1.2). More precisely, we will
prove the following theorem.
Theorem C. (Existence, uniqueness, and stability of positive stationary solutions)
(1) If (H1)- (H3) hold, then (1.2) has exactly two stationary solutions in X+p , u 0,
which is linearly unstable, and u+( )2X+p nf0g, which is globally asymptotically with
respect to perturbations in X+p nf0g.
(2) If a> 0 and (H1)-(H2) are satis ed, where a := 1p1p2 p
N
R
Df(x;0)dx with D = [0;p1]
[0;p2] [0;pN]. then (H3) is satis ed and the conclusions in (1) hold.
Let
u+inf = inf
x2RN
u+(x): (2.23)
The following four theorems are about the spatial spreading speeds of (1.2).
Theorem D. (Existence and symmetry of spreading speeds) Assume (H1) - (H3).
(1) The spreading speed c ( ) of (1.2) in the direction of 2SN 1 exists for every 2SN 1
and
c ( ) = inf >0 0( ; ) ;
where 0( ; ) is the principal spectrum point of (2.8) with a(x) = f(x;0).
(2) Assume that k(z) = k( z) for z2RN. c ( ) = c ( ) for every 2SN 1.
(3) For every u0 2X+( ) and cc ( ),
lim sup
t!1
sup
x ct
u(t;x;u0;z) = 0 uniformly in z2RN:
Theorem E. (Spreading features of spreading speeds) Assume (H1) - (H3).
(1) If u0 2 X+ satis es that u0(x) = 0 for x 2 RN with jx j 1, then for each
c>maxfc ( );c ( )g,
lim sup
t!1
sup
jx j ct
u(t;x;u0;z) = 0 uniformly in z2RN:
(2) Assume that 2 SN 1 and 0 < c < minfc ( );c ( )g. Then for any > 0, and
r> 0,
lim inft!1 inf
jx j ct
(u(t;x;u0;z) u+(x+z)) = 0 uniformly in z2RN
for every u0 2X+ satisfying u0(x) for all x2RN with jx j r.
Theorem F. (Spreading features of spreading speeds) Assume (H1) - (H3).
(1) If u0 2X+ satis es that u0(x) = 0 for x2RN with kxk 1, then
lim sup
t!1
sup
kxk ct
u(t;x;u0;z) = 0 uniformly in z2RN:
for all c> sup
2SN 1
c ( ).
(2) Assume that 0 0, there is r> 0 such that
lim inft!1 inf
kxk ct
(u(t;x;u0;z) u+(x+z)) = 0 uniformly in z2RN
for every u0 2X+ satisfying u0(x) for x2RN with kxk r.
19
To indicate the dependence of the spreading speeds on the growth rate function f,
denote the spreading speed c ( ;f). If no confusion exists, we still use c ( ).
Let f(u) = 1jDjRDf(x;u)dx; with D =
NY
i=1
[0;pi] andjDj=
NY
i=1
pi where the period vector
p = (p1;:::;pN).
Theorem G. (E ect of spatial variation) Assuming that f(0) > 0, c ( ;f) c ( ; f) for
any 2 SN 1. Moreover, assuming also (H4), c ( ;f) = c ( ; f) for some 2 SN 1 i
f(x;0) f(0) is independent of x.
Theorems D-G extend the spreading speed theory for (1.3) to (1.2) and establish some
fundamental theories for the further study of the spreading and propagating dynamics of
(1.2). The next natural and important problems to address include the existence, uniqueness,
and stability of traveling wave solutions of (1.2) in the direction of connecting u+ and u
with speed c c ( ). To explore this, we assume the existence of the principal eigenvalue of
(2.8).
We now state the main results of the dissertation on traveling wave solutions. For given
2SN 1 and c>c ( ), let 2(0; ( )) be such that
c = 0( ; ;a0) :
and ( ) is such that
c ( ) = 0( ;
( );a0)
( ) :
If (H4) holds, let ( )2X+p be the positive principal eigenfunction ofK ; I +a0( )I with
k ( )kXp = 1.
Theorem H. (Existence of traveling wave solutions) Assume (H1)-(H4). Then for any
2SN 1 and c > c ( ), there is a bounded measurable function : RN RN !R+ such
that the following hold.
(1) ( ; ) generates a traveling wave solution connecting u+( ) and 0 and propagating in the
direction of with speed c. Moreover, lim
x !1
(x;z)
e x (x+z) = 1 uniformly in z2R
N.
20
(2) Let U(t;x;z) = u(t;x; ( ;z);z)(= (x ct ;z +ct )). Then
Ut(t;x;z) > 0 8t2R; x;z2RN;
lim
x ct! 1
Ut(t;x;z) = 0, and lim
x ct!1
Ut(t;x;z)
e (x ct) (x+z) = c uniformly in z2R
N.
Remark 2.2. Let (x;z) be as in Theorem H and ( ;z) = ( ;z ). Then U(t;x;z) =
(x ct;z +x) and ( ;z) is di erentiable in and ( ;z) < 0.
Theorem I. (Uniqueness and continuity of traveling wave solutions) Assume the same con-
ditions as in Theorem H. Let ( ; ) be as in Theorem H.
(1) Suppose that 1( ; ) also generates a traveling wave solution of (1.2) in the direction of
with speed c and lim
x !1
1(x;z)
(x;z) = 1 uniformly in z2R: Then 1(x;z) (x;z):
(2) (x;z) is continuous in (x;z)2RN.
Theorem J. (Stability of traveling wave solutions) Assume the same conditions as in The-
orem H.
Let U(t;x) = U(t;x; 0) = (x ct ;ct ), where ( ; ) is as in Theorem H. For any u0 2X+
satisfying that lim
x !1
u0(x)
U(0;x) = 1 and lim infx ! 1u0(x) > 0, there holds
limt!1 sup
x2RN
u(t;x;u0;0)U(t;x) 1
= 0:
We remark that by the spreading property of c ( ), it is not di cult to see that (1.2)
has no traveling wave solutions in the direction of 2SN 1 with propagating speed smaller
than c ( ). Theorems H-J show the existence, uniqueness, and stability of traveling wave
solutions of (1.2) in any given direction with speed greater than the spreading speed in that
direction for the following three special but important cases, that is, the nonlocal dispersal is
nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous
in a region where it is most conducive to population growth in the zero-limit population. It
should be pointed out that in the last case, for N = 1;2, (H4) is automatically satis ed.
21
It remains open whether a general spatially periodic monostable equation with nonlocal
dispersal in RN with N 3 has traveling wave solutions connecting the spatially periodic
positive stationary solution u+ and 0 and propagating with constant speeds.
22
Chapter 3
Comparison Principle and Sub- and Super-solutions
In this chapter, we establish some basic properties of solutions of equation (1.2) and
some related nonlocal linear evolution equations, including the comparison principle and
monotonicity of solutions with respect to initial conditions, convergence of solutions on
compact sets.
3.1 Solutions of Evolution Equation and Semigroup Theory
For given 2SN 1, 2R, and a( )2Xp, consider also
@u
@t =
Z
RN
e (y x) k(y x)u(t;y)dy u(t;x) +a(x)u(t;x); x2RN: (3.1)
Let X and Xp be as in (2.2) and (2.4), respectively. For given 0, let
X( ) =fu2C(RN;R)jthe function x7!e kxku(x) belongs to Xg (3.2)
equipped with the norm kukX( ) = sup
x2RN
e kxkju(x)j. Note that X(0) = X.
It follows from the general linear semigroup theory (see [27] or [50]) that for every u0 2
X( ) ( 0), (3.1) has a unique solution u(t; ;u0; ; )2X( ) with u(0;x;u0; ; ) = u0(x).
Put
(t; ; )u0 = u(t; ;u0; ; ): (3.3)
Note that for every 2R and 0, there is !( ; ) > 0 such that
k (t; ; )u0kX( ) e!( ; )tku0kX( ) 8t 0; 2SN 1; u0 2X( ): (3.4)
23
Note also that if u0 2Xp, then (t; ; )u0 2Xp for t 0.
By general nonlinear semigroup theory (see [27] or [50]), (1.2) has a unique (local)
solution u(t;x;u0) with u(0;x;u0) = u0(x) for every u0 2 X. Also if u0 2 Xp, then
u(t;x;u0)2Xp for t in the existence interval of the solution u(t;x;u0).
Due to the spatial inhomogeneity of (1.2), it is sometime important to consider the
space shifted equation (2.15) of (1.2) and the following space shifted equation of (3.1),
@u
@t =
Z
RN
e (y x) k(y x)u(t;y)dy u(t;x) +a(x+z)u(t;x); (3.5)
where z2RN. Note that if = 0 and a(x) = a0(x)(:= f(x;0)), then (3.5) reduces to the
space shifted equation of the linearization equation (2.10) of (1.2) at u = 0,
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +a0(x+z)u(t;x); x2RN: (3.6)
It is again a consequence of the general semigroup theory that (2.15) has a unique (local)
solution u(t;x;u0;z) with u(0;x;u0;z) = u0(x) (z2RN) for every u0 2X. Also given u0 2
X( ) ( 0), (3.5) has a unique solution u(t;x;u0; ; ;z) with u(0;x;u0; ; ;z) = u0(x).
We set
(t; ; ;z)u0 = u(t; ;u0; ; ;z): (3.7)
Sometimes we need study the solutions on the space ~X, where ~X is as (2.13). For
example, to get a continuous solution of (2.15), we may rst investigate the existence of
solution with u0 2 ~X and then prove the continuity. It is again a consequence of the general
semigroup theory that (2.15) has a unique (local) solution u(t;x;u0;z) with u(0;x;u0;z) =
u0(x) (z2RN) for every u0 2 ~X.
Throughout this chapter, we assume that 2SN 1 and 2R are xed, unless otherwise
speci ed.
24
3.2 Sub- and Super-solutions
Let X+p and X+ be as in (2.5) and (2.3), respectively. Let
Int(X+p ) =fv2Xpjv(x) > 0;x2RNg: (3.8)
For v1;v2 2Xp, we de ne
v1 v2 (v1 v2) if v2 v1 2X+p (v1 v2 2X+p );
and
v1 v2 (v1 v2) if v2 v1 2Int(X+p ) (v1 v2 2Int(X+p )):
For u1;u2 2X, we de ne
u1 u2 (u1 u2) if u2 u1 2X+ (u1 u2 2X+):
De nition 3.1. A bounded Lebesgue measurable function u(t;x) on [0;T) RN is called a
super-solution (or sub-solution) of (2.15) if for any x2RN, u(t,x) is absolutely continuous
on [0;T)(and so @u@t exists a.e on [0,T)) and satis es that for each x2RN,
@u
@t (or )
Z
RN
k(y x)u(t;y)dy u(t;x) +f(x+z;u)u(t;x)
for a.e. t2(0;T).
Sub and super-solutions of (3.5) are de ned similarly.
3.3 Comparison Principle and Monotonicity
Proposition 3.1 (Comparison principle).
25
(1) If u1(t;x) and u2(t;x) are sub-solution and super-solution of (3.1) on [0;T), respec-
tively, u1(0; ) u2(0; ), and u2(t;x) u1(t;x) 0 for (t;x) 2 [0;T) RN and
some 0 > 0, then
u1(t; ) u2(t; ) for t2[0;T):
(2) If u1(t;x) and u2(t;x) are bounded sub- and super-solutions of (1.2) on [0;T), respec-
tively, and u1(0; ) u2(0; ), then u1(t; ) u2(t; ) for t2[0;T).
(3) For every u0 2X+, u(t;x;u0) exists for all t 0.
Proof. (1) We prove the proposition by modifying the arguments of [35, Proposition 2.4].
First let v(t;x) = ect u2(t;x) u1(t;x) . Then v(t;x) satis es
@v
@t
Z
RN
e (y x) k(y x)v(t;y)dy +p(x)v(t;x); x2RN (3.9)
for t2(0;T), where p(x) = a(x) 1 +c. Take c> 0 such that p(x) > 0 for all x2RN. We
claim that v(t;x) 0 for t2[0;T) and x2RN.
Let p0 = sup
x2RN
p(x). It su ces to prove the claim for t2 (0;T0) and x2 RN, where
T0 = minfT; 1p0+Mg with M = RRN e z k(z)dz. Assume that there are ~t 2 (0;T0) and
~x2RN such that v(~t; ~x) < 0. Then there is t0 2(0;T0) such that
vinf := inf
(t;x)2[0;t0] RN
v(t;x) < 0:
Observe that there are tn2(0;t0] and xn2RN such that
v(tn;xn)!vinf as n!1:
26
By (3.9), we have
v(tn;xn) v(0;xn)
Z tn
0
Z
RN
e (y xn) k(y xn)v(t;y)dy +p(xn)v(t;xn) dt
Z tn
0
Z
RN
e (y xn) k(y xn)vinfdy +p0vinf dt
= tn(M +p0)vinf
t0(M +p0)vinf
for n = 1;2; . Note that v(0;xn) 0 for n = 1;2; . We then have
v(tn;xn) t0(M +p0)vinf
for n = 1;2; . Letting n!1, we get
vinf t0(M +p0)vinf >vinf (since 0 vinf < 0):
This is a contradiction. Hence v(t;x) 0 for (t;x)2[0;T) RN and then u1(t;x) u2(t;x)
for (t;x)2[0;T) RN.
(2) Let v(t;x) = ect(u2(t;x) u1(t;x)). Then v(t; ) 0 and v(t;x) satis es
@v
@t
Z
RN
k(y x)v(t;y)dy +p(t;x)v(t;x); x2RN
for t2(0;T), where
p(t;x) = c 1 +f(x;u2(t;x)) + u1(t;x)
Z 1
0
fu(x;su1(t;x) + (1 s)u2(t;x))ds v(t;x)
for t2[0;T), x2RN. By the boundedness of u1(t;x) and u2(t;x), there is c> 0 such that
inf
t2[0;T);x2RN
p(t;x) > 0:
27
(2) then follows from the arguments in (1) with p(x) and p0 being replaced by p(t;x) and
sup
t2[0;T);x2RN
p(t;x), respectively.
(3) By (H1), there is M > 0 such that
u0(x) M and f(x;M) < 0 for x2RN:
Let uM(t;x) M for x2RN and t2R. Then uM is a super-solution of (1.2) on [0;1).
Let I(u0) R be the maximal interval of existence of the solution u(t; ;u0) of (1.2). Then
by (2), one obtains
0 u(t;x;u0) M for x2RN; t2I(u0)\[0;1):
It then follows easily that [0;1) I(u0) and u(t;x;u0) exists for all t 0.
The following remark follows by the arguments similar to those in Proposition 3.1 (1).
Remark 3.1. Suppose that u1;u2 2X( ) and u1 u2. Then (t; ;0;0)u1 (t; ;0;0)u2
for all t> 0.
Proposition 3.2 (Strong monotonicity). Suppose that u1;u2 2Xp and u1 u2, u1 6= u2.
(1) (t; ; )u1 (t; ; )u2 for all t> 0.
(2) u(t; ;u1) u(t; ;u2) for every t> 0 at which both u(t; ;u1) and u(t; ;u2) exist.
Proof. (1) We apply the arguments in Theorem 2.1 of [37]. First, assume that u0 2X+p nf0g.
Then by Proposition 3.1 (1), (t; ; )u0 0 for t> 0.
We claim that eK ; tu0 0 for t> 0. In fact, note that
eK ; tu0 = u0 +tK ; u0 + t
2(K ; )2u0
2! +:::+
tn(K ; )nu0
n! +:::
28
Let x0 2 RN be such that u0(x0) > 0. Then there is r > 0 such that u0(x0) > 0 for
x2B(x0;r) :=fy2RNjky x0k 0 for x2B(x0;r + ):
By induction,
(K ; u0)n(x) > 0 for x2B(x0;r +n ); n = 1;2; :
This together with the periodicity of u0(x) implies that eK ; tu0 0 for t> 0.
Let m> 1 min
x2RN
a(x). Note that
(t; ; )u0 = u(t; ;u0) = e(K ; I+a( )I+mI mI)tu0 = e mIte(K ; I+a( )I+mI)tu0
and (e mItv)(x) = e mtv(x) for every x2RN. Note also that
e(K ; I+a( )I+mI)tu0 = eK ; tu0 +
Z t
0
eK ; (t s)( I +a( )I +mI)u(s; ;u0)ds for t> 0:
It then follows that (t; ; )u0 0 for all t> 0.
Now let u0 = u2 u1. Then u0 2X+p nf0g. Hence (t; ; )u0 0 for t > 0 and
then (t; ; )u1 (t; ; )u2 for t> 0.
(2) Let v(t;x) = u(t;x;u2) u(t;x;u1) for t 0 at which both u(t;x;u1) and u(t;x;u2)
exist. Then v(0; ) = u2 u1 0 and v(t;x) satis es
@v
@t =
Z
RN
k(y x)v(t;y)dy v(t;x) +f(x;u(t;x;u2))v(t;x)
+ u(t;x;u1)
Z 1
0
fu(x;su(t;x;u1) + (1 s)u(t;x;u2))ds v(t;x); x2RN:
(2) then follows from the arguments similar to those in (1).
29
3.4 Convergence on Compact Sets
In this section, we investigate the convergence of solutions of (1.2) or (3.1) on compact
sets.
First, we prove the following lemma.
Lemma 3.1. For given 0 0 and fung2X( 0) with kunkX( 0) M for some M > 0 and
n = 1;2; , un(x)!0 as n!1 uniformly for x in bounded subsets of RN if and only if
un(x)!0 in X( ) as n!1 for every > 0.
Proof. Suppose that un(x)!0 as n!1 uniformly for x in bounded subsets of RN, that
is, for any > 0 and L > 0, there exists N0 2N such that jun(x)j< for all n > N0 and
kxk L. For given > 0, pick ^ 2 ( 0; ). Note that kunkX( 0) M for some M > 0
and n = 1;2; . Then for any > 0, there exists an L> 0 such that je ^ kxkun(x)j< for
kxk> L and n = 1;2; . It then follows that for any > 0, there is N0 2N such that
je ^ kxkun(x)j< for all n>N0 and x2RN. This implies that un!0 in X( ) as n!1
for every > 0.
Suppose that un(x) ! 0 in X( 0) as n!1 for some 0 > 0, that is, kunkX( 0) ! 0
as n!1. For any > 0, L > 0, let 0 = e 0L, then there exists a N0 2N such that
je 0kxkun(x)j< 0 for x2RN and n N0, which implies that jun(x)j 0 and n = 1;2; ,
and un(x) ! u0(x) as n ! 1 uniformly for x in bounded subsets of RN, then
(t; ; )u
n
(x)! (t; ; )u
0
(x) as n!1 uniformly for (t;x) in bounded subsets
of [0;1) RN.
(2) If un 2 X+ and u0 2 X+ are such that kunkX M for some M > 0 and n =
1;2; and un(x) ! u0(x) as n !1 uniformly for x in bounded subsets of RN,
30
then u(t;x;un) ! u(t;x;u0) as n ! 1 uniformly for (t;x) in bounded subsets of
[0;1) RN.
Proof. (1) First of all, by Lemma 3.1, for every given > 0, kun u0kX( ) !0 as n!1.
By (3.4),
k (t; ; )un (t; ; )u0kX( ) e!( ; )tkun u0kX( )
for t 0 and n = 1;2; . Then, k (t; ; )un (t; ; )u0kX( ) !0 as n!1 uniformly
for t in bounded subsets of [0;1). This implies that
(t; ; )u
n
(x)! (t; ; )u
0
(x)
as n!1 uniformly for (t;x) in bounded subsets of [0;1) RN.
(2) By Proposition 3.1 (3), u(t;x;u0) and u(t;x;un) exist on [0;1) for any n 1 and
there is M > 0 such that ku(t; ;u0)kX M and ku(t; ;un)kX M for t 0 and
n = 1;2; . Let vn(t;x) = u(t;x;un) u(t;x;u0) for x2RN, t 0, and n = 1;2; . Then
@vn
@t =
Z
RN
k(y x)vn(t;y)dy vn(t;x) +an(t;x)vn(t;x); x2RN;
where
an(t;x) = f(x;u(t;x;un)) + u(t;x;u0)
Z 1
0
fu(x;su(t;x;u0) + (1 s)u(t;x;un))ds vn(t;x):
Hence
vn(t; ) = e(K I)tvn(0; ) +
Z t
0
e(K ; I)(t s)an(s; )vn(s; )ds:
Note that for every > 0 there are !0( )2R and L0 > 0 such that
ke(K I)tvkX( ) e!0( )tkvkX( ) 8v2X( )
31
and
jan(t;x)j L0 8t 0; x2RN:
It then follows from Gronwall?s inequality that
kvn(t; )kX( ) e(!0( )+L0)tkvn(0; )kX( ):
By the arguments in (1), we have vn(t;x)!0 and hence u(t;x;un)!u(t;x;u0) as n!1
uniformly for (t;x) in bounded subsets of [0;1) RN.
32
Chapter 4
Spectral Theory of Dispersal Operators
In this chapter, we investigate the eigenvalue problem (2.8) and prove Theorems A and
B stated in the chapter 2 and some other related results which are used in the proof of the
existence of spreading speeds of (1.2) in later chapters. The results of this chapter in the
case that the nonlocal kernel function has compact support have already been published (see
[56], [57]).
Throughout this chapter, Xp is as in (2.4), a : RN !RN is a smooth function, a2Xp,
and
amax = max
x2RN
a(x); amin = min
x2RN
a(x):
a( )I : Xp ! Xp has the same meaning as in (2.7) with a0( ) being replaced by a( ) and
K ; : Xp !Xp is understood as in (2.9), 2SN 1, and 2R. We rst introduce in 4.1
some important operators related toK ; I+a( )I or (2.8) and explore some basic properties
of the eigenvalue problems associated with these operators. We then prove Theorems A in
4.2 and derive in 4.3 from Theorems A some results on the spectral radius of some operator
related to K ; I +a( )I.
4.1 Evolution Operators and Eigenvalue Problems
In this section, we introduce some evolution operators related to the operatorK ; I+
a( )I, explore the basic properties of the eigenvalue problems associated to these operators,
and discuss the relations between the eigenvalues ofK ; I+a( )I and its related operators.
If no confusion occurs, we may write the principal eigenvalue ( ; ;a) ofK ; I +a( )I (if
exists) as ( ; ).
33
First of all, we introduce a compact operator associated toK ; based on the perturba-
tion idea in [7]. This operator plays an important role in the proofs of Theorems A in the
next section. For given > 1 +amax, let U ; ; : Xp!Xp be de ned as follows
(U ; ; u)(x) =
Z
RN
e (y x) k(y x)u(y)
+ 1 a(y) dy: (4.1)
Observe that U ; ; is a compact and positive operator on Xp. Let r(U ; ; ) be the spectral
radius of U ; ; .
Proposition 4.1. (1) > 1 +amax is an eigenvalue of K ; I +a( )I or (2.8) i 1 is
an eigenvalue of the eigenvalue problem
U ; ; v = v:
(2) For > 1 + amax, 1 is an eigenvalue of U ; ; with a positive eigenfunction i
r(U ; ; ) = 1.
(3) If there is > 1+amax with r(U ; ; ) > 1, then there is 0 > such that r(U ; ; 0) =
1.
(4) If > 1 + amax is an eigenvalue of K ; I + a( )I or (2.8) with a positive eigen-
function, then it is the principal eigenvalue of (2.8).
Proof. (1) and (2) follow from Proposition 2.1 of [7].
(3) and (4) follow from Theorem 2.2 of [7].
By Proposition 4.1, the spectral radius of U ; ; provides a useful tool for the investiga-
tion of those eigenvalues ofK ; I +a( )I which are greater than 1 +amax. The following
proposition shows that if K ; I + a( )I possesses a principal eigenvalue, then it must be
greater than 1 +amax.
34
Proposition 4.2. If ( ; ) is the principal eigenvalue of K ; I + a( )I, then ( ; ) >
1 +amax.
Proof. Since ( ; ) is the principal eigenvalue of K ; I +a( )I, there is an eigenfunction
2X+p nf0g such that
Z
RN
e (y x) k(y x) (y)dy (x) +a(x) (x) = ( ; ) (x); x2RN: (4.2)
Note that u(t;x) = e ( ; )t (x) is a solution of (3.1). By Proposition 3.2, 2Int(X+p ).
Let x0 2RN be such that a(x0) = amax. By 2Int(X+p ),
Z
RN
e (y x0) k(y x0) (y)dy> 0:
This together with (4.2) implies that
( ; ) (x0) > (x0) +a(x0) (x0):
Hence ( ; ) > 1 +amax.
Next, consider the evolution equation (3.1) associated with the operatorK ; I+a( )I.
Let (t; ; ) be the solution operator of (3.1) given in (3.3) and p(t; ; ) : Xp !Xp be
de ned by
p(t; ; ) = (t; ; )jXp (4.3)
for t 0, 2SN 1 and 2R. Let r( p(1; ; )) and ( p(1; ; )) be the spectral radius
and the spectrum of p(1; ; ), respectively. The following lemma states the relationship
between the principal eigenvalue ofK ; I+a( )I and the spectral radius of p(1; ; ) and
follows easily (see [27, Theorems 1.5.2 and 1.5.3]).
Lemma 4.1. The principal eigenvalue ( ; ) of (2.8) exists if and only if r( p(1; ; )) is
an algebraically simple eigenvalue of p(1; ; ) with an eigenfunction in X+p and for every
35
~ 2 p(1; ; ) nfr( p(1; ; ))g, j~ j< r( p(1; ; )). Moreover, if ( ; ) exists, then
( ; ) = lnr( p(1; ; )) .
Therefore, the spectral radius of p(1; ; ) plays an important role in the investigation
of the principal eigenvalue of K ; I + a( )I or (2.8). We next establish some further
observations for r( p(1; ; )).
Note that (t; ;0) is independent of 2SN 1. We put
~ (t) = (t; ;0) (4.4)
for 2SN 1.
For given u0 2X and 2R, letting u ; 0 (x) = e x u0(x), then u ; 0 2X(j j). The
following lemma follows directly from the uniqueness of solutions of (3.1).
Lemma 4.2. For given u0 2X, 2SN 1, and 2R, (t; ; )u0 = e x ~ (t)u ; 0 .
Observe that for each x2RN, there is a measure m(x;y;dy) such that
(~ (1)u0)(x) =
Z
RN
u0(y)m(x;y;dy): (4.5)
Moreover, by (~ (1)u0( piei))(x) = (~ (1)u0( ))(x piei) for x2RN and i = 1;2; ;N,
Z
RN
u0(y)m(x piei;y;dy) =
Z
RN
u0(y piei)m(x;y;dy) =
Z
RN
u0(y)m(x;y +piei;dy)
and hence
m(x piei;y;dy) = m(x;y +piei;dy) (4.6)
for i = 1;2; ;N.
By Lemma 4.2, we have
( (1; ; )u0)(x) =
Z
RN
e (x y) u0(y)m(x;y;dy); u0 2X:
36
Proposition 4.3. For every u2Int(X+p ),
inf
x2RN
R
RN e
(x y) u(y)m(x;y;dy)
u(x) r(
p(1; ; ))
sup
x2RN
R
RN e
(x y) u(y)m(x;y;dy)
u(x) :
Proof. By [20, Theorems 3.6 and 4.3], the spectral radius of the nonnegative operator
p(1; ; ) is bounded by the lower and upper Collatz-Wielandt numbers of u for every
u2 Int(X+p ), which are de ned by supf 0 : u p(1; ; )ug and inff 0 : u
p(1; ; )ug, respectively. The inequality then follows.
In proving the existence of spreading speeds of (1.2) in chapter 6, properly truncated
operators of (1; ; ) are used. We therefore introduce them next.
Let : R![0;1] be a smooth function satisfying that
(s) =
8
>><
>>:
1 for jsj 1
0 for jsj 2:
(4.7)
For a given B > 0, de ne B(1; ; ) : X!X by
( B(1; ; )u0)(x) =
Z
RN
e (x y) u0(y) (ky xk=B)m(x;y;dy): (4.8)
De ne pB(1; ; ) : Xp!Xp by
pB(1; ; ) = B(1; ; )jXp: (4.9)
Similarly, let r( pB(1; ; )) and ( pB(1; ; )) be the spectral radius and the spectrum of
pB(1; ; ), respectively.
Lemma 4.3.
k pB(1; ; ) p(1; ; )kXp !0 as B!1
37
uniformly for in bounded sets and 2SN 1.
Proof. It su ces to prove that
Z
ky xk B
e ky xkm(x;y;dy)!0 as B!1
uniformly for in bounded sets and for x2RN.
For given 0 > 0 and n2N, let un2X( 0 + 1) be such that
un(x) =
8
>><
>>:
e 0kxk for kxk n
0 for kxk n 1
and
0 un(x) e 0n for kxk n:
ThenkunkX( 0+1) !0 as n!1. Therefore,k~ (1)unkX( 0+1) !0 as n!1: This together
with Lemma 3.1 implies that
Z
RN
un(y)m(x;y;dy)!0 as n!1
uniformly for x in bounded subsets of RN and then
Z
kyk n
e 0kykm(x;y;dy)!0 as n!1
uniformly for x in bounded subsets of RN. The later implies that
Z
ky xk n
e ky xkm(x;y;dy)!0 as n!1
38
uniformly for j j 0 and x in bounded subset of RN. By (4.6), for every 1 i N,
Z
ky (x+piei)k n
e ky (x+piei)km(x+piei;y;dy) =
Z
ky xk n
e ky xkm(x+piei;y +piei;dy)
=
Z
ky xk n
e ky xkm(x;y;dy):
We then have Z
ky xk n
e ky xkm(x;y;dy)!0 as n!1
uniformly for j j 0 and x2RN. The lemma now follows.
4.2 Existence of the Principal Eigenvalue
In this section, we prove Theorems A. Throughout this section, U ; ;a is understood as
in (4.1), and r(U ; ;a) denotes the spectral radius of U ; ;a. We may simply write U for U ; ;a
if no confusion can occur.
Proof of Theorem A. (1) We prove the existence of a 0 > 0 and the existence of a principal
eigenvalue ( ; ;a) for all 0 < < 0, 2SN 1 and 2R. By Proposition 4.1, it su ces
to prove the existence of 0 > 0 such that for each 0 < < 0, 2SN 1, and 2R, there
exists an > 1 +amax such that r(U ) > 1.
Let
M0 = inf
2SN 1
2(amax amin + 1)
R
z >0(z )
2k(z)dz
1=2
:
We rst prove the existence of an > 1 + amax such that r(U ) > 1 for every 2SN 1,
> 0, and 2R with j j> M0 .
39
In fact, for v(x) 1 and every 0 < < 1 and > 0, we have
(U 1+amax+ v)(x) =
Z
RN
e (y x) k(y x)
amax + a(y) dy
Z
RN
e (y x) k(y x)
amax amin + dy
Z
z <0
e z k(z)
amax amin + dz
1a
max amin +
1 + 2 2
2!
Z
z <0
(z )2k(z)dz
+
4 4
4!
Z
z <0
(z )4k(z)dz +
2 2
M20 : (4.10)
Similarly, we have (U 1+amax+ v)(x) 2 2M2
0
for < 0. Hence if j j > M0, then for
0 < 1, there is > 1 such that
(U 1+amax+ v)(x) > v(x) 8x2RN:
This implies that r(U 1+amax+ ) > 1.
We then only need to prove that there is a 0 > 0 and an > 1 +amax with r(U ) > 1
for all 0 < < 0, 2SN 1, and 2R with M0. We prove this by applying arguments
similar to those in [37, Theorem 2.6].
Let D = [0;p1] [0;p2] [0;pN]. Assume that x0 2D is such that a(x0) = amax.
Without loss of generality, we may assume that x0 2Int(D).
Then for every 0 < < 1, there is some > 0 such that a(x0) a(x) < for x 2
B( ;x0) D, where B( ;x0) = fx 2 RNjkx x0k < g. Let v( ) 2 Xp be such that
40
v(x) = (kx x0k) for x2D, where
(r) =
8
>>>>
<
>>>
>:
cos( r2 ) if 0 r
0 if r>
Let 0 < < 2 and 0 < 1 < 1. Also let D1 = B( 2;x0) , D2 = B( ;x0)nD1. For x2D2, let
~D( ;x) = B( ;x)\B(kx x0k;x0).
Observe that for x2B( 2;x0), v(x)
p2
2 . For x2D2 and y2 ~D( ;x), v(y) v(x).
For x2DnB( ;x0), v(x) = 0. Observe also that there are C > 0 (independent of ) and
1 > 0 such that
inf
x2D1
Z
B( =2;x0)
e (y x) k(y x)dy C; inf
x2D2
Z
~D( ;x)
e (y x) k(y x)dy C
for 0 < < 1, 2SN 1, and 0 j j M0.
Clearly, for each > 1,
(Uamax 1v)(x) v(x) for x2DnB( ;x0): (4.11)
If x2D1, we have
(Uamax 1v)(x)
Z
D
e (y x) k(y x)v(y)
1 a(y) +amax 1 dy
11
1 +
Z
B( ;x0)
e (y x) k(y x)v(y)dy
p2
2(1 1 + )
Z
B( =2;x0)
e (y x) k(y x)dy
p2C
2(1 1 + )
p2C
2(1 1 + )v(x): (4.12)
41
If x2D2, we have
(Uamax 1v)(x)
Z
D
e (y x) k(y x)v(y)
1 a(y) +amax 1 dy
11
1 +
Z
D
e (y x) k(y x)v(y)dy
v(x)1
1 +
Z
~D( ;x)
e (y x) k(y x)dy
Cv(x)1
1 +
: (4.13)
Let M =
p2C
2(1 1+ ). By (4.11)-(4.13) and the periodicity of v, we obtain
(Uamax 1v)(x) Mv(x) for all x2RN:
Choose and 1 such that 0 < <
p2
2 C and 1 > 1 > 1 +
p2C
2 . Let 0 = minf 1;
2g.
Then M > 1 and r(Uamax 1) M > 1. Thus (1) is proved.
(2)
By the arguments in (4.10), we have for v(x) 1 and every 0 < < 1 that
(U 1+amax+ v)(x) =
Z
RN
e (y x) k(y x)
amax + a(y) dy
a
max amin +
for 2SN 1, and 2R. Hence if amax amin < , then for 0 < 1, there is > 1 such
that
(U 1+amax+ v)(x) > v(x):
This implies that r(U 1+amax+ ) > 1. It then follows from Proposition 4.1 that the principal
eigenvalue ( ; ;a) of (2.8) exists for 2SN 1 and 2R.
(3)
42
Let x0 2D be such that a(x0) = amax. Also, without loss of generality, we may assume
that x0 2 Int(D). Since the partial derivatives of a(x) up to order N 1 at x0 are zero,
there is M > 0 such that
a(x0) a(y) Mkx0 ykN for y2RN: (4.14)
Fix 2 SN 1, and 2 R. Let > 0 be such that < 2 and B(2 ;x0) D. Let
v 2X+p be such that v (x) = 1 if x2B( ;x0) and v (x) = 0 if x2DnB(2 ;x0).
Clearly, for every x2DnB(2 ;x0) and > 1,
(U 1+amax+ v )(x) > v (x) = 0: (4.15)
For x2B(2 ;x0), there is ~M > 0 such that
e (y x) k(y x) ~M
for y2B( ;x0). It then follows that for x2B(2 ;x0)
(U 1+amax+ v )(x)
Z
B( ;x0)
e (y x) k(y x)
Mkx0 ykN + dy
Z
B( ;x0)
~M
Mkx0 ykN + dy: (4.16)
Note that RB( ;x0) ~MMkx0 ykNdy =1. This together with the periodicity of v (x) implies
that for 0 < 1, there is > 1 such that
(U 1+amax+ v )(x) > v (x) for x2RN: (4.17)
Hence r(U 1+amax+ ) > 1. It then follows from Proposition 4.1 that the principal eigenvalue
( ; ;a) of (2.8) exists for 2SN 1 and 2R.
43
Next we prove a proposition about the comparison of principal eigenvalue on the a( )
of (2.8), which will also be used in the later chapters.
Proposition 4.4. Assume that a1(x) ~a1(x). If for given 2 R, and 2 SN 1, both,
( ; ;a1) and ( ; ;~a1), exist, then
( ; ;a1) ( ; ;~a1):
Proof. Consider the following two evolution equations,
@u
@t =
Z
RN
e (y x) k(y x)u(t;y)dy u(t;x) +a1(x)u(t;x) (4.18)
and
@u
@t =
Z
RN
e (y x) k(y x)u(t;y)dy u(t;x) + ~a1(x)u(t;x): (4.19)
For given u0 2 Xp, let u(t; ;u0) and ~u(t; ;u0) be the solutions of (4.18) and (4.19) with
u(0; ;u0) = u0 and ~u(0; ;u0) = u0, respectively. Put
(t)u0 = u(t; ;u0); ~ (t)u0 = ~u(t; ;u0):
Let r( (1)) and r(~ (1)) be the spectral radius of (1) and ~ (1), respectively. By [56,
Lemma 3.1],
( ; ;a1) = lnr( (1))
and
( ; ;~a1) = lnr(~ (1)):
By the fact that a1 ~a1 and Proposition 3.1, we have that for any u0 0, (t; ;u0)
~ (t; ;u0) for any t 0. It then follows that r( (1)) r(~ (1)) and then ( ; ;a1)
( ; ;~a1).
44
Theorem 4.1. (1) For each 2SN 1, ( ; ;a) is convex in ;
(2) There are m> 0 and 0 > 0 such that ( ; ;a) m 2 for all 0 and 2SN 1;
(3) If ( ;0;a) > 0, then for every 2SN 1, there is a ( )2(0;1) such that
( ; ( );a)
( ) = inf >0
( ; ;a)
(4.20)
and
( ; ;a)
>
( ; ( );a)
( ) for 0 < <
( ): (4.21)
Proof. (1) Fix 2 SN 1. By Lemma 4.1, ^ ( i) := r( p(1; ; i)) is an eigenvalue of
p(1; ; i) with a positive eigenfunction i (i = 1;2). Hence
^ ( i) = ( p(1; ; i) i)(x)
i(x) =
R
RN e
i(x y) i(y)m(x;y;dy)
i(x) 8x2R
N
for i = 1;2. For given 0 t 1, let 3 = t1 1 t2 . By H older?s inequality,
[^ ( 1)]t[^ ( 2)]1 t = [
R
RN e
1(x y) 1(y)m(x;y;dy)
1(x) ]
t[
R
RN e
2(x y) 2(y)m(x;y;dy)
2(x) ]
1 t
Z
RN
[e
1(x y) 1(y)
1(x) ]
t[e
2(x y) 2(y)
2(x) ]
1 tm(x;y;dy)
=
R
RN e
(t 1+(1 t) 2)(x y) 3(y)m(x;y;dy)
3(x) 8x2R
N:
Applying Proposition 4.3, we get
[^ ( 1)]t[^ ( 2)]1 t sup
x2RN
R
RN e
(t 1+(1 t) 2)(x y) 3(y)m(x;y;dy)
3(x) r(
p(1; ;t 1 +(1 t) 2):
Thus,
ln[^ ( 1)]t[^ ( 2)]1 t ln(r( p(1; ;t 1 + (1 t) 2)):
45
By Lemma 4.1 again,
t ( ; 1;a) + (1 t) ( ; 2;a) ( ;t 1 + (1 t) 2;a);
that is, ( ; ;a) is convex in .
(2) Note that by Proposition 4.4 ( ; ;a) ( ; ;amin), and
( ; ;amin) =
Z
RN
e y k(y)dy 1 +amin
with 1 as an eigenfunction. Let mn( ) = Ry <0 ( y )nn! k(y)dy. Then, for > 0
Z
RN
e y k(y)dy 1 +amin =
1X
n=0
Z
RN
( y )n
n! k(y)dy 1 +amin
m2( ) 2 +
1X
n=2
m2n( ) 2n +amin
Let m := inf
2SN 1
m2( )(> 0) and 0 > 0 be such that P1n=2m2n( ) 2n >jaminj for 0.
Then m and 0 have the required property.
(3) By (2), ( ; ;a) !1 as !1. By ( ;0;a) > 0, ( ; ;a) !1 as ! 0+. This
implies that there is ( ) > 0 such that (4.20) and (4.21) hold.
4.3 Spectral Radius of the Truncated Evolution Operator
In this section, we derive from Theorems A and B some important properties of the
spectral radius of the truncated operator pB(1; ; ) of p(1; ; ) discussed in 3.1.
For a xed 2 SN 1, let rB( ; ) = r( pB(1; ; )) and B( ; ) = lnr( pB(1; ; )).
Denote by 0B( ; ) the partial derivative of B( ; ) with respect to . We establish the
following theorem for B( ; ), which is analogous to Theorem A for ( ; ).
46
Theorem 4.2. Let 2 SN 1 be given. Assume that (2.8) has the principal eigenvalue
( ; ) for 2R, that ( ;0) > 0, and that ( ; ( )) ( ) < ( ; ( )+k0) ( )+k0 for some k0 > 0, where
( ) is as in Theorem A. Then we have:
(1) There is B0 > 0 such that for each B B0 and j j ( ) + k0, r( pB(1; ; )) is an
algebraically simple eigenvalue of pB(1; ; ) with a positive eigenfunction. Moreover,
B( ;0) > 0 and B( ; ( )) ( ) < B( ; ( )+k0) ( )+k0 .
(2) For each B B0, lnr( pB(1; ; )) (i.e. B( ; )) is convex in for j j ( ) +k0.
(3) For a given B B0, de ne
B( ) := inf ~ B( ) B( ; ~
B( ))
~ B( ) = inf0< ( )+k0
B( ; )
:
Then (i) B( ) > 0 and 0B( ; ) < B( ; ) for 0 < < B( ):
(ii) For every > 0, there exists some > 0 such that for < < B( ),
0B( ; ) < B( ;
B( ))
B( ) + :
(iii) For every > 0, there is B1 B0 such that if B also satis es B B1, then
j ( ;
( ))
( )
B( ; B( ))
B( ) j< :
Proof. (1) It follows from Lemma 4.3 and the perturbation theory of the spectrum of bounded
linear operators (see [38]).
(2) It follows from arguments similar to those in Theorem 4.1.
(3) Fixing 2 SN 1, we set B( ) = B( ; ) and rB( ) = rB( ; ) for simplifying
notations. By (1), 0 < B( ) < ( ) +k0.
47
For 0 < ( ) +k0, let
( ) = B( ) and ( ) = ( ( ))0 0B( ) = r
0
B( )
rB( ):
By the convexity of B( ) in 2( ( ) k0; ( ) + k0), 0 0 for 0 < < ( ) + k0.
Note that
0( ) = 1 [ ( ) ( )]; (4.22)
and
( 2 0)0 = 0( ) 0 (4.23)
for 0 < < ( ) +k0.
By (4.22) and 0( B( )) = 0, we have
0B( B( )) = ( B( )) = ( B( )) = B(
B( ))
B( ) :
By the de nition of B( ),
B( )
>
B( B( ))
B( ) for 2(0;
B( )):
By 00B( ) 0( ) 0 for 2(0; ( ) +k0),
0B( ) 0B( B( )) for 2(0; B( )):
It then follows that
0B( ) < B( ) for 2(0; B( )):
(i) is thus proved.
48
By the continuity of 0B( ), for every > 0, there is > 0 such that
0B( B( )) 0B( ) < for 2( ; B( )):
This together with 0B( B( )) = B( B( ))
B( )
implies that
0B( ) < B(
B( ))
B( ) + for 2( ;
B( )):
Hence (ii) holds.
Note that for every 0 < 1, there are 0 < ~ < ( ) < ~ + < ( ) +k0 such that
( ; ~ )
~ =
( ; ~ + )
~ + =
( ; ( ))
( ) +
2
( ; )
( ; ( ))
( ) for 2[~
; ~
+
]:
Note also that there is B1 B0 such that if B B1, then
( ; )
B( )
<
4 for 2[~
; ~
+
]
holds. This implies that
B( ( ))
( ) < minf
B(~ )
~ ;
B(~ + )
~ + g:
By (4.23) and 0( B( )) = 0,
0( ) 0 for 2(0; B( )) and 0( ) 0 for 2( B( ); ( ) +k0):
We thus must have
B( )2[~ ; ~ + ]
49
and then
j ( ;
( ))
( )
B( B( ))
B( ) j j
( ; ( ))
( )
( B( ))
B( ) j+j
( B( ))
B( )
B( B( ))
B( ) j< ;
i.e. (iii) holds.
4.4 Remarks
It is sometime important to study the space shifted equation (3.5) of (3.1). Let (t; ; ;z)
be the solution operator of (3.5) given in (3.7) and
p(t; ; ;z) = (t; ; ;z)jXp: (4.24)
Note that (t; ;0;z) is independent of 2SN 1. Put
~ (t;z) = (t; ;0;z) (4.25)
for 2SN 1.
The following remarks are easy to derive.
Remark 4.1. For every z 2 RN, ( (t; ; ;z)u0)(x) = ( (t; ; )u0( z))(x + z) and
(~ (t;z)u0)(x) = (~ (t)u0( z))(x+z).
Remark 4.2. For every z2RN,
(~ (1;z)u0)(x) =
Z
RN
u0(y z)m(x+z;y;dy) (4.26)
and
( (1; ; ;z)u0)(x) =
Z
RN
e (x+z y) u0(y z)m(x+z;y;dy): (4.27)
50
Remark 4.3. If the principal eigenvalue ( ; ) of (2.8) exists and (x; ; ) is a correspond-
ing eigenfunction, then for every z2RN, = ( ; ) is an eigenvalue and (x; ; ;z) :=
(x+z; ; ) is a corresponding eigenfunction of the following space shifted eigenvalue prob-
lem of (2.8),
8
>><
>>:
R
RN e
(y x) k(y x)v(y)dy v(x) +a(x+z)v(x) = v; x2RN
v(x+piei) = v(x); i = 1;2; ;N; x2RN:
(4.28)
Let : R ! [0;1] be a smooth function satisfying (4.7). For a given B > 0, de ne
B(1; ; ;z) : X!X by
( B(1; ; ;z)u0)(x) =
Z
RN
e (x+z y) u0(y z) (ky x zk=B)m(x+z;y;dy) (4.29)
Let
pB(1; ; ;z) = B(1; ; ;z)jXp: (4.30)
Remark 4.4. It follows from the arguments of Lemma 4.3 that
k pB(1; ; ;z) p(1; ; ;z)kXp !0 as B!1
uniformly for in bounded sets and z2[0;p1] [0;p2] [0;pN].
Remark 4.5. The spectral radius r( pB(1; ; ;z)) of pB(1; ; ;z) is independent of z 2
RN. If r( pB(1; ; )) is an eigenvalue and B(x; ; ) is a corresponding eigenfunction of
pB(1; ; ), then r( pB(1; ; ;z))(= r( pB(1; ; ))) is an eigenvalue of pB(1; ; ;z) with
the eigenfunction B(x+z; ; ).
51
4.5 An Example
In this section, we give an example which shows that the principal eigenvalue of (2.8)
may not exist in case that N 3.
Example. Let q(x) be a smooth p-periodic function de ned as follows
q(x) =
8
>><
>>:
e
kxk2
kxk2 2 for kxk<
0 for kxk 1=2;
where p = (1;1; ;1) (i.e. q(x + ei) = q(x) for i = 1;2; ;N) and 0 < < 1=2. Note
that qmax = 1, q(x) decreases as kxk increases and q(x) e kxk
2
2 for kxk 1=2. Let
k(z) = 1 N ~k(z ), where ~k( ) be as in (2.1). Then for given M > 1, K ; I + QM or (2.8)
with = 0 and a(x) = Mq(x) has no principal eigenvalue for 0 < 1 and 1, where
QM = Mq( )I.
In fact, let UM = U ;0; , where U ;0; is as in (4.1) with = 0 and a( ) = Mq( ). If is
the principal eigenvalue of K ; I + QM, then by Propositions 4.1 and 4.2, > 1 + M
and r(UM ) = 1. Observe that for every > 0 and u(x) 1,
(UM 1+M+ 1)(x) =
Z
RN
k(y x)
+M Mq(y)dy
1 +M
Z
RN
k(y x)dy +CN
( )
N
Z
kyk
1
+M 1 e kyk
2
2
dy
1 +M +CN
( ) N
N
Z
kyk 1
1
+M 1 e kyk2 dy;
where N ( ) is the total number of disjoint unit hypercubes in RN whose vertices have integer
coordinates and lie inside the ball B(x; +pN) =fy2RNjky xk +pNg. We then
have N ( ) = O( N) as !1 and hence
N ( )
N = O(1) as !1:
52
Note that when N 3, there is ~M such that
Z
kyk 1
1
+M 1 e kyk2 dy
~M
M
for all > 0 and M > 0. This implies that for every > 0 and M > 0,
(UM 1+M+ 1)(x) 1 +M +CN
( ) N
N
~M
M:
Therefore when N 3, there is 0 < 0 < 1 such that
UM 1+M+ 1 1 0
for 0 < < 0, 1 and any > 0. It then follows that
r(UM 1+M+ ) 1 0
and hence r(UM ) 1 0, a contradiction. Therefore, for the given M > 1 and 0 < 1,
K ; I +QM has no principal eigenvalue for 1.
We remark that the principal eigenvalue ofK ; I +QM (if exists) depends on the
parameters , M, , and N. To see the dependence of (if exists) on M, x N 3, > 0,
and 0 < < 1=2 such that
C ~MN
( ) N
N < 1:
Let (M) = (if exists) and M be the corresponding positive eigenfunction with
k M( )kXp = 1. By Theorem B (1), if 0 < M < 1, then (M) exists. By the above
arguments, (M) does not exists for M 1. We claim that there is M > 1 such that
(M) exists for 0 1 such that
r(UM 1+M)
8
>>>
>>><
>>>
>>>:
> 1 for 0 M :
We then have that for 0 1+M, and the principal eigenvalue ofK ; I+QM does not exists for M M .
Moreover, it is clear that lim
M!M
r(UM 1+M) = 1 and hence (4.31) holds. Note that for every
0 0,
0( ; ;a ) 0( ; ;an) 0( ; ;a) 0( ; ;an) 0( ; ;a+ ) 8n 1: (5.3)
This together with 0( ; ;a ) = 0( ; ;a) implies (5.2).
Lemma 5.2. Given a2Xp, 0( ; ;a) 0( ; ; a) for any 2SN 1 and 2R.
Proof. Take an2CN(RN)\Xp such that an satis es (H5) and
an( ) a( ) for n 1 and kan akXp !0 as n!1:
By Theorem A, ( ; ;an) exists and ( ; ;an) = 0( ; ;an) for n 1. By Theorem
B, 0( ; ;an) 0( ; ; an) for n 1. The lemma follows by letting n!1 and applying
Lemma 5.1.
To prove the theorem, we will apply Proposition 3.1 and so rst we provide a sub-solution
and a super-solution of (2.15).
Proposition 5.1. Assume (H4) and let be the positive principal eigenfunction of K I +
a( )I with k kXp = 1. Then for any z2RN and 0 < b 1, ^v(t;x;z;b) := b (x + z) is a
sub-solution of (2.15).
Proof. Fix z2RN. Observe that
59
Z
RN
k(y x) (y +z)dy 0(x+z) +f(x+z;0) (x+z) = 0 (x+z) 8x2RN:
Observe also that max
x2RN
0 (x+z) > 0 and then
0b (x+z) (f(x+z;0) f(x+z;b (x+z)))b (x+z) 80 ** 0) and an is such that (H5) holds.
Proof. This follows by arguments similar to those in Proposition 5.1.
Proposition 5.3. For d 1, z2RN, v(t;x;z) d is a super-solution of (2.15).
Proof. By direct calculation, we have
@ v
@t [
Z
RN
k(y x) v(t;y;z)dy v(t;x;z) +f(x+z; v) v(t;x;z)]
d[
Z
RN
k(y x)dy 1 +f(x+z;d)]
0:
The proposition thus follows.
Lemma 5.3. Assume (H1) and (H2). (1.2) has at most one positive stationary solution u+( )
in X+p . If there is a positive stationary solution u+( ) 2 X+p , it is globally asymptotically
stable with respect to perturbations in X+p .
60
Proof. It follows from the arguments in [37, Lemma 3.3].
Proof of Theorem C. (1) It follows easily that u(t; ; v) is monotonically decreasing as t in-
creasing, where v is as in Proposition 5.3. Let u+(x) = limt!1u(t;x; v). Then u+(x) is upper
semicontinuous and satis ed that RRN k(y x)u+(y)dy u+(x) + f(x + z;u+(x))u+(x) =
0. Then u+(x)[1 f(x + z;u+(x))] = RRN k(y x)u+(y)dy > 0, which implies that
f(x+z;u+(x)) < 1.
Let g(x) = RRN k(y x)u+(y)dy and y = u+(x). Let F(x;z;y) = g(x) y+f(x+z;y)y
and then F(x;z;y) = 0. Since @F(x;z;y)@y = 1 + f(x + z;y) + fu(x + z;y)y < 0, by Implicit
Function Theorem, u+(x) is continuous.
Similarly, let u (x) = limt!1u(t;x;v) and then is also a positive stationary solution of
(2.15). By Lemma 5.3, u (x) = u+(x)
For any u0 2X+p nf0g, for t0 > 0, there exist v and v such that v > u(t0;x;u0) > v,
where v is as in Proposition 5.2. By Proposition 3.1, u(t;x; v) >u(t+t0;x;u0) >u(t;x;v).
Then, u = u+ is a globally asymptotically stable stationary solutions with respect to
the perturbations in X+p nf0g. (1) then follows.
(2) By Lemma 5.2, we have 0(a) 0( a) = a>0 and then (H3) is satis ed. Thus the
conclusions in (1) hold.
Remark 5.1. Assume (H1), (H2), and (H3). Then
limt!1(u(t;x; +;z) u+(x+z)) = 0
holds uniformly in x2RN and z2RN for every + > 0. Here + in u(t;x; +;z) stands
for the constant function with value +.
61
Chapter 6
Spreading Speeds of Spatially Periodic Nonlocal Monostable Equations
In this chapter, we investigate the spatial spreading speeds of (1.2) and prove Theorems
D-G. To do so, we rst introduce a so-called spreading speed interval [c inf( );c sup( )] of (1.2)
in the direction of 2SN 1 and establish basic properties. We will prove the existence of
spreading speed of (1.2) in the direction of 2SN 1 by showing that [c inf( );c sup( )] is a
singleton and c inf( )(= c sup( )) is the spreading speed of (1.2) in the direction of . The
results of this chapter in the case that the nonlocal kernel function has compact support
have been published (see [29], [56], [57]).
6.1 Spreading Speed Intervals
Throughout this section, Xp is as in (2.4), X is as in (2.2), and X+( ) is as in (2.12)
( 2SN 1). We assume (H1) - (H3). and so, (1.2) has a unique positive stable periodic
equilibrium solution u+(x) in Xp. Let u+inf be as in (2.23). For simplifying notations set
lim inf
x ! 1
u0(x) = limr! 1 inf
x r
u0(x); lim sup
x !1
u0(x) = limr!1 sup
x r
u0(x)
for given u0 2X and 2SN 1. For given u(t; )2X, 2SN 1, and c2R, put
lim inf
x ct;t!1
u(t;x) = lim inft!1 inf
x ct
u(t;x); lim sup
x ct;t!1
u(t;x) = lim sup
t!1
sup
x ct
u(t;x);
lim inf
jx j ct;t!1
u(t;x) = lim inft!1 inf
jx j ct
u(t;x); lim sup
jx j ct;t!1
u(t;x) = lim sup
t!1
sup
jx j ct
u(t;x);
62
and
lim inf
kxk ct;t!1
u(t;x) = lim inft!1 inf
kxk ct
u(t;x); lim sup
kxk ct;t!1
u(t;x) = lim sup
t!1
sup
kxk ct
u(t;x):
De nition 6.1. For a given vector 2SN 1, let
C inf( ) =
n
cj8u0 2X+( ); lim inf
x ct;t!1
(u(t;x;u0) u+(x)) = 0
o
and
C sup( ) =
n
cj8u0 2X+( ); lim sup
x ct;t!1
u(t;x;u0) = 0
o
:
De ne
c inf( ) = supfcjc2C inf( )g; c sup( ) = inf fcjc2C sup( )g:
We call [c inf( );c sup( )] the spreading speed interval of (1.2) in the direction of .
Observe that if c1 2C inf( ) and c2 2C sup( ), then c1 < c2. Hence c inf( ) c sup( ) for
all 2SN 1.
To establish basic properties of the spreading speed intervals of (1.2), we rst construct
some useful sub- and super-solutions of (1.2) and its space shifted equation (2.15). Recall
that u(t;x;u0;z) denotes the solution of (2.15) with u(0;x;u0;z) = u0(x) for u0 2X and
z2RN.
Let (s) be the function de ned by
(s) = 12(1 + tanh s2); s2R: (6.1)
Observe that
0(s) = (s)(1 (s)); s2R (6.2)
63
and
00(s) = (s)(1 (s))(1 2 (s)); s2R: (6.3)
Without loss of generality, we may assume that f(x;u) = 0 for u 0. For otherwise,
let ~ ( ) 2C1(R) be such that ~ (u) = 1 for u 0 and ~ (u) = 0 for u 0. We replace
f(x;u) by f(x;u)~ (u). Hence we may also assume that there is u < 0 such that for any
u0 2X with u u0 0 and z2RN,
u u(t; ;u0;z) 0 for t 0: (6.4)
Proposition 6.1. Assume (H1) - (H3). Let (u 0 < + 2u+inf) be given
constants. There is C0 > 0 such that for every C C0, every 2SN 1 and every z2RN,
the following properties hold:
1) letting v (t;x;z) = u(t;x; ;z) (x +Ct) +u(t;x; ;z)(1 (x +Ct)), v+ and
v are super- and sub-solutions of (2.15) on [0;1), respectively;
2) letting w (t;x;z) = u(t;x; ;z) (x Ct) +u(t;x; ;z)(1 (x Ct)), w+ and
w are super- and sub-solutions of (2.15) on [0;1), respectively.
Proof. We prove that v+(t;x;z) with z = 0 is a super-solution of (1.2). Other statements
can be proved similarly. We write v+(t;x) for v+(t;x; 0).
64
First, by Taylor expansion,
f(x;u(t;x; +)) (x +Ct) +f(x;u(t;x; ))(1 (x +Ct))
f x;u(t;x; +) (x +Ct) +u(t;x; )(1 (x +Ct)
= f(x;u(t;x; +) u(t;x; ) +u(t;x; )) (x +Ct)
+f(x;u(t;x; ))(1 (x +Ct))
f(x;(u(t;x; +) u(t;x; )) (x +Ct) +u(t;x; ))
=
fu(x; ~u (t;x) +u(t;x; )) fu(x; ~u (t;x) (x +Ct) +u(t;x; ))
(u(t;x; +) u(t;x; )) (x +Ct)
= fuu(x;u (t;x)) u (t;x) u(t;x; ) u(t;x; +) u(t;x; ) 0(x +Ct)
where u (t;x) = ~u (t;x) + u(t;x; ) and u (t;x) and u (t;x) are between u(t;x; ) and
u(t;x; +). Then a direct computation yields
v+t (t;x) [
Z
RN
k(y x)v+(t;y)dy v+(t;x)] f(x;v+(t;x))
= 0(x +Ct)
n
C u(t;x; +) u(t;x; )
Z
RN
k(y x)(u(t;y; +) u(t;y; )) (y +Ct) (x +Ct) 0(x +Ct) dy
fuu(x;u (t;x)) u (t;x) u(t;x; ) (u(t;x; +) u(t;x; ))
o
:
Note that there are M0 and M1 > 0 such that
u(t;x; +) u(t;x; ) M0 for all t 0; x2RN;
j (y +Ct) (x +Ct) 0(x +Ct) j M1 for all t 0; x;y2RN; ky xk :
65
It then follows that there is C0 > 0 such that for every C C0, v+(t;x) is a super-solution
of (1.2).
Proposition 6.2. Assume (H1) - (H3). For every 2SN 1, the following properties hold:
(1) if there is u 0 2X+( ) such that
lim inf
x ct;t!1
u(t;x;u
0;z) u
+(x+z) = 0 uniformly in z2RN;
then c c inf( );
(2) if cc sup( ), then for every u0 2X+( ),
lim sup
x ct;t!1
u(t;x;u0;z) = 0 uniformly in z2RN:
Proof. It can be proved by arguments similar to those in [30, Lemma 3.5].
Corollary 6.1. Assume (H1) - (H3). [c inf( );c sup( )] is a nite interval for all 2SN 1.
66
Proof. Fix 2SN 1.
Let = 0 < + u+inf be given constants. There is u 0 2X+( ) such that
w+(0;x;z) = (x ) + +(1 (x )) u 0(x); x2RN
for all z2RN. Then by Propositions 3.1 and 6.1,
w+(t;x;z) = u(t;x; ;z) (x C0t) +u(t;x; +;z)(1 (x C0t))
u(t;x;z;u 0)
for t 0, and x;z2RN. This implies that for C >C0,
lim sup
x Ct;t!1
u(t;x;z;u 0) = 0 uniformly in z2RN:
Therefore by Proposition 6.3, c sup( ) C0.
Now let u+inf > + > 0 > u be a given constant, where u satis es (6.4). There
is u 0 2X+( ) such that
v (0;x;z) = (x ) + +(1 (x )) u 0 (x)
for x;z2RN. Then by Propositions 3.1 and 6.1 again,
v (t;x;z) = u(t;x; ;z) (x +C0t) +u(t;x; +;z)(1 (x +C0t))
u(t;;x;u 0 ;z)
for t 0, and x;z2RN. This implies that for C < C0,
lim inf
x Ct;t!1
(u(t;x;u0;z) u+(x+z)) = 0 uniformly in z2RN:
67
Therefore by Proposition 6.2, c inf( ) C0.
Hence [c inf( );c sup( )] is a nite interval.
Let
~X+( ) =fu2X+j lim inf
x ! 1
u0(x) > 0;lim sup
x !1
u0(x) = 0g: (6.5)
Proposition 6.4. Assume (H1) - (H3).
(1) Let 2SN 1, u0 2 ~X+( ), and c2R be given. If there are 0 and T0 > 0 such that
lim inf
x cnT0;n!1
u(nT0;x;u0;z) 0 uniformly in z2RN; (6.6)
then for every c0 0 such that
lim inf
jx j cnT0;n!1
u(nT;x;u0;z) 0 uniformly in z2RN; (6.7)
then for every c0 0 such that
lim inf
kxk cnT0;n!1
u(nT;x;u0;z) 0 uniformly in z2RN; (6.8)
68
then for every c0 0, there exists n1 n0 such that
u(t;x; ~u0;z) u+(x+z) for t n1T0; x;z2RN: (6.10)
For a given B > 1, let ~uB( ) 2X be such that 0 ~uB(x) 02 for x2RN, ~uB(x) = 02 for
x B 1, and ~u0(x) = 0 for x B. By Proposition 3.3, Remark 5.1 and (6.10), there
is ~B0 > 1 such that for each B ~B0,
u(t;0; ~uB;z) u+(z) 2 for n1T0 t (n1 + 1)T0; z2RN: (6.11)
Note that (c c0)nT0 !1 as n!1. Hence there is n2 n1 such that
(c c0)nT0 ~B0 +c0(n1 + 1)T0 for n n2:
This together with (6.9) implies that
u(nT0; +x+c0nT0 +c0 ;u0;z) ~u~B0( )
69
for all x2RN with x 0, all with n1T0 (n1 + 1)T0, and all n n2. Given n n2
and (n+n1)T0 t< (n+n1 + 1)T0, let = t nT0. Then n1T0 < (n1 + 1)T0 and
u(t;x+c0t ;u0;z) = u( ;x+c0t ;u(nT0; ;u0;z);z)
= u( ;0;u(nT0; +x+c0nT0 +c0 ;u0;z);z +x+c0t )
u+(x+z +c0t ) 2
for all x2RN with x 0. It then follows that
u(t;x;u0;z) u+(x+z) 2 for z2RN; x c0t; t (n1 +n2)T0:
(1) is thus proved.
(2) It can be proved by arguments similar to those in (1).
(3) It can also be proved by arguments similar to those in (1). For the reader?s conve-
nience, we provide a proof in the following.
First, for a given c0 0, there is n1 n0 such that
u(t;x; ~u0;z) u+(x+z) for t n1T0; x;z2RN: (6.13)
For a given B > 1, let ~uB( ) 2X be such that 0 ~uB(x) 02 for x2RN, ~uB(x) = 02 for
kxk B 1, and ~u0(x) = 0 forkxk B. By Proposition 3.3, Remark 5.1, and (6.12), there
exists ~B0 > 1 such that
u(t;0; ~uB;z) u+(z) 2 for n1T0 t (n1 + 1)T0; z2RN (6.14)
70
for all B ~B0.
Note that (c c0)nT0 !1 as n!1. Hence there is n2 n1 such that
(c c0)nT0 ~B0 +c0(n1 + 1)T0 for n n2:
This together with (6.12) implies that
u(nT0; +x;u0;z) ~u~B0( )
for each n n2 and each x2RN with kxk c0nT0 + c0(n1 + 1)T0. For given n n2 and
(n+n1)T0 t< (n+n1 + 1)T0, let = t nT0. Then n1T0 < (n1 + 1)T0 and
u(t;x;u0;z) = u( ;x;u(nT0; ;u0;z);z)
= u( ;0;u(nT0; +x;u0;z);z +x)
u+(x+z) 2
for all x2RN with kxk c0t( c0(n+n1 + 1)T0). This implies that
u(t;x;u0;z) u+(x+z) 2
for t (n1 +n2)T0 and kxk c0t. (3) is thus proved.
6.2 Spreading Speeds under the Assumption of the Existence of a Principal
Eigenvalue
In this section, we investigate the spreading speeds of (1.2) and prove Theorems D, E
and F stated in the chapter 2 under the assumptions (H1)-(H4).
Recall that u(t;x;u0) denotes the solution of (1.2) with u(0; ;u0) = u0 2 X and
u(t;x;u0;z) denotes the solution of (2.15) withu(0; ;u0;z) = u0 2X. Note thatu(t;x;u0;0) =
71
u(t;x;u0). In the following, (t; ; ;z) and B(1; ; ;z) denote the solution operators of
(3.5) with a(x) = a0(x)(= f(x;0)) given in (3.7) and the truncated operator of (1; ; ;z)
given in (4.29), respectively.
Proof of Theorem D. (1) Fix 2 SN 1. Put ( ) = ( ; ). By Theorem 4.1, there is
= ( )2(0;1) such that
inf
>0
( )
=
( )
:
It is easy to see that c ( ) exists and c ( ) = ( ) if and only if c inf( ) = c sup( ) = ( ) .
We rst prove that c sup( ) ( ) .
Since f(x;u) = f(x;0) + fu(x; )u for some 0 u, we have, by assumption (H2),
f(x;u) f(x;0) for u 0. If u0 2X+ , then
u(t;x;u0) ( (t; ;0;0)u0)(x) for x2RN: (6.15)
Suppose that ( ;x) 2 X+p is a principal eigenvector of (2.8) with a(x) = a0(x)(=
f(x;0), that is, (K ; I + a0( )I) ( ;x) = ( ) ( ;x) with > 0. It can easily be
veri ed that ( (t; ;0;0)~u0)(x) = Me (x ~ct) ( ;x) with ~u0 = Me x ( ;x) for ~c =
( )
and M > 0. For any u0 2 X
+( ), choose M > 0 large enough such that ~u0
u0. Then by Proposition 3.1 and Remark 3.1 we have u(t;x;u0) ( (t; ;0;0)u0)(x)
( (t; ;0;0)~u0)(x) = Me (x ~ct) ( ;x). Hence
lim sup
x ct;t!1
u(t;x;u0) = 0 for every c> ~c:
This implies that c sup( ) ( ) for any > 0 and then
c sup( ) inf >0 ( ) : (6.16)
72
We then prove that c inf( ) inf >0 ( ) . We will do so by modifying the arguments in
[44] and [62].
First of all, for every 0 > 0, there is b0 > 0 such that
f(x;u) f(x;0) 0 for 0 u b0; x2RN: (6.17)
Choose B 1 such that Theorem 4.2 holds. Observe that if u0 2 X+ is so small that
0 u(t;x;u0;z) b0 for t2[0;1], x2RN and z2RN, then
u(1;x;u0;z) e 0( (1; ;0;z)u0)(x) e 0( B(1; ;0;z)u0)(x) (6.18)
for x2RN and z2RN.
Let rB( ) be the spectral radius of B(1; ; ;0) and B( ) = lnrB( ). By Theorem
4.2 (1), rB( ) is an eigenvalue of B(1; ; ;0) with a positive eigenfunction ( ;x) for
j j ( ) +k0.
By Theorem 4.2 (3), for each 1 > 0, there exists a B > 0 such that
B(
B( ))
B( )
( ( ))
( ) + 1; (6.19)
where B( ) is as in Theorem 4.2 (3). Moreover, there is 1 such that
0B( ) < B(
B( ))
B( ) + 1 (6.20)
for 1 < < B( ). In the following, we x 2 ( 1; B( )). By Theorem 4.2 (3) again,
we can choose 0 > 0 so small that
B( ) r0B( )=rB( ) 0 > 0: (6.21)
73
Let
( ;z) = ( ;z) ( ;z) :
For given > 0 and z2RN, de ne
( ;z) = 1 tan 1
R
RN ( ;y)e
(y z) (ky zk=B) sin ( (y z) + ( ;y))m(z;y;dy)
R
RN ( ;y)e
(y z) (ky zk=B) cos ( (y z) + ( ;y))m(z;y;dy):
It is not di cult to prove that
lim !0 ( ;z) = 0B( ) + ( ;z) uniformly in z2RN:
Choose > 0 so small that (B +j (z)j+j ( ;z)j) < for all z2RN and
( ;z) ( ;z) < 0B( ) + 1 (6.22)
for z2[0;p1] [0;p2] [0;pN].
For given 2 > 0 and > 0, de ne
v(s;x) =
8
>><
>>:
2 ( ;x)e s sin (s ( ;x)); 0 s ( ;x)
0; otherwise:
(6.23)
Let
v (x;s;z) = v(x +s ( ;z) + ( ;z);x+z):
Choose 2 > 0 so small that
0 u(t;x;v ( ;s;z);z) b0 for t2[0;1]; x;z2RN:
Let
( ; ;z) = ( ;z) + ( ;z):
74
Then for 0 s ( ;z) , we have
u(1;0;v ( ;s;z);z)
e 0 B(1; ;0;z)v ( ;s;z)
2e 0
Z
RN
h
( ;y)e [(y z) +s+ ( ; ;z)] sin [(y z) +s+ ( ; ;z) ( ;y)]
(ky zk=B)
i
m(z;y;dy)
= e 0v(s;z)e ( ; ;z) sec ( ;z) ( ;z)
Z
RN
h
( ;y)e (y z) cos ( (y z) + ( ;y))
(ky zk=B)
i
m(z;y;dy):
Observe that
lim !0e 0e ( ; ;z) sec ( ;z) ( ;z)
Z
RN
h
( ;y)e (y z) cos ( (y z) + ( ;y))
(ky zk=B)
i
m(z;y;dy)
= e 0e r0B( )=rB( )rB( )
= e B( ) r0B( )=rB( ) 0
> 1 (by (6.21)):
It then follows that for 0 s ( ;z) ,
u(1;0;v ( ;s;z);z) v(s;z) = v (( ( ;z) ( ;z)) ;s;( k( ;z) + ( ;z)) +z):
Clearly, the above equality holds for all s2R.
Let s(x) be such that v( s(x);x) = maxs2Rv(s;x). Let
v(s;x) =
8>
><
>>:
v( s(x);x); s s(x)
v(s+ ;x); s s(x) :
75
Set
v (x;s;z) = v(x +s ( ;z) + ( ;z);x+z):
We then have
u(1;0; v ( ;s;z);z) v(s;z) = v (( ( ;z) ( ;z)) ;s;( ( ;z) + ( ;z)) +z)
for s2R and z2RN.
Let
v0(x;z) = v(x ;x+z):
Note that v(s;x) is non-increasing in s. Hence we have
u(1;x;v0( ;z);z) = u(1;0;v0( +x;z);x+z)
= u(1;0; v ( ;x + ( ;x+z) ( ;x+z);x+z);x+z)
v(x + ( ;x+z) ( ;x+z);x+z)
v(x 0B( ) + 1;x+z) (by (6.22))
v
x B(
B( ))
B( ) + 2 1;x+z
(by (6.20))
v
x (
( ))
( ) + 3 1;x+z
(by (6.19))
= v0
x [ (
( ))
( ) 3 1)] ;[
( ( ))
( ) 3 1] +z
for z2[0;p1] [0;p2] [0;pN]. Let ~c ( ) = ( ( )) ( ) 3 1. Then
u(1;x;v0( ;z);z) v0(x ~c ( ) ;~c ( ) +z)
76
for all z2RN. We also have
u(2;x;v0( ;z);z) u(1;x;v0( ~c ( ) ;~c ( ) +z);z)
= u(1;x ~c ( ) ;v0( ;~c ( ) +z);~c ( ) +z)
v0(x 2~c ( ) ;2~c ( ) +z)
for all z2RN. By induction, we have
u(n;x;v0( ;z);z) v0(x n~c ( ) ;n~c ( ) +z)
for n 1 and z2RN. This together with Proposition 6.4 implies that
c ( ) ~c ( ) = (
( ))
( ) 3 1:
Since 1 is arbitrary, we must have
c inf( ) inf >0 ( ) : (6.24)
By (6.16) and (6.24), we have c ( ) exists and c ( ) = inf >0 ( ) .
(2) LetDi = [i1p1;(i1+1)p1] [i2p2;(i2+1)p2] ::: [iNpN;(iN+1)pN](i = (i1;i2; ;iN)2
ZN). Let 1 = ( ; ) and 2 = ( ; ) be the principal eigenvalues ofK ; I +a( )I and
K ; I+a( )I with eigenfunctions 1; 2 2Int(X+p ), respectively. It su ces to prove that
1 = 2. Observe that
Z
RN
e (y x) k(y x) 1(y)dy 1(x) +a(x) 1(x) = 1 1(x); x2RN
and Z
RN
e (y x) k(y x) 2(y)dy 2(x) +a(x) 2(x) = 2 2(x); x2RN:
77
Multiplying the rst equality by 2(x) and the second one by 1(x) and then integrating
both equations over D0, we get
Z
D0
[
Z
RN
e (y x) k(y x) 1(y)dy 2(x) 1(x) 2(x)+a(x) 1(x) 2(x)]dx = 1
Z
D0
1(x) 2(x)dx
and
Z
D0
[
Z
RN
e (y x) k(y x) 2(y)dy 1(x) 2(x) 1(x)+a(x) 2(x) 1(x)]dx = 2
Z
D0
2(x) 1(x)dx:
Therefore, in order to derive 1 = 2, we only need to prove
Z
D0
Z
RN
e (y x) k(y x) 1(y) 2(x)dydx =
Z
D0
Z
RN
e (y x) k(y x) 2(y) 1(x)dydx:
To this end, it su ces to prove that for each i = (i1;i2; ;iN)2ZN, one has
Z
D0
Z
Di
e (y x) k(y x) 1(y) 2(x)dydx =
Z
D0
Z
D i
e (y x) k(y x) 2(y) 1(x)dydx:
For given i = (i1;i2; ;iN)2ZN, let zl = yl ilpl and wl = xl ilpl, for l = 1;2;:::;N. We
have
Z
D0
Z
Di
e (y x) k(y x) 1(y) 2(x)dydx
=
Z
Di
Z
D0
e (y x) k(y x) 1(y) 2(x)dxdy
=
Z
D0
Z
D i
e (z w) k(z w) 1(z1 +i1p1;:::;zN +iNpN) 2(w1 +i1p1;:::;wN +iNpN)dwdz
=
Z
D0
Z
D i
e (w z) k(z w) 2(w) 1(z)dwdz:
This proves (2).
(3) It follows from (1) and Proposition 6.2 (2).
(4) It follows from (1) and Proposition 6.3 (2).
78
Proof of Theorem E. (1) For given u0 in (1), there are u 0 2X+( ) such that
u0( ) u 0 ( ):
By Proposition 6.3, for every c>c ( ),
lim sup
x ct;t!1
u(t;x;u+0 ;z) = 0; lim sup
x ( ) ct;t!1
u(t;x;u 0 ;z) = 0
uniformly in z2RN. By Proposition 3.1 and Proposition 3.2,
u(t;x;u0;z) u(t;x;u 0 ;z) for t 0; x;z2RN:
It then follows that
lim sup
jx j ct;t!1
u(t;x;u0;z) = 0 uniformly in z2RN:
(2) First, we claim that for each > 0, there is r > 0 such that
lim inft!1 inf
jx j ct
(u(t;x;u0) u+(x)) = 0 (6.25)
for every u0 2X+ satisfying u0(x) for all x2RN with jx j r . By Proposition 3.1
and Proposition 3.2, we only need to consider satisfying 0 < **__><
>>:
; r 0
0; r 1:
Let
u ; (x) = ~u (x ( )):
79
By the de nition of c ( ),
lim inf
x ( ) ct;t!1
(u(t;x;u ; ;z) u+(x+z)) = 0 uniformly in z2RN:
Take any 0 < ~c 0, let ~u B2C(R;R) be such that ~u B(r) 0 and
~u B(r) =
8
>><
>>:
~u (r); B r
0; r B 1:
Let
u ; B (x) = ~u B(x ( )):
Then
u(t;x;u ; B ;z)!u(t;x;u ; ;z)
as B!1 in open compact topology. This implies that there are T > 1c ~c and B0 > 0 such
that given B B0,
u(T;x;u ; B ;z)
for 0 x ( ) cT, kxk 2cT, and z 2 RN. Note that for each x 2 RN with
0 x ( ) cT, there is a vector q such that q = 0 andk(x q)k 2cT. It then follows
that
u(T;x;u ; B ;z) = u(T;x q;u ; B ;z +q)
for 0 x ( ) cT, and z2RN.
Let r > 0 be such that r > B0 + 1. Assume that u0 0 satis es u0(x) for
jx j r . Then
u0( r ) u ; B ( ) for all r with 0 r r 1:
80
It then follows from the above arguments that
u(T;x;u0;z) for r cT + 1 x r +cT 1
for all z2RN. This together with T > 1c ~c implies that
u(T;x;u0;z) for jx j r + ~cT:
By induction, we have
u(nT;x;u0;z) for jx j r + ~cnT; n = 1;2; :
Then by Proposition 6.4, one obtains for each 0 0 and r > 0 be given. Suppose that u0 2X+ satis es u0(x) for
all x2RN with jx j r. Note that there is m> 0 such that
1 +f(x;u(t;x;u0)) m 8x2RN; t 0:
Then
ut(t;x;u0)
Z
RN
k(y x)u(t;y;u0)dy mu(t;x;u0)
81
and hence
(emtu(t;x;u0))t
Z
RN
k(y x)emtu(t;y;u0)dy:
This together with Proposition 3.1 implies that
emtu(t; ;u0) eKtu0
where eKt = I + Kt + K2t22! + and Ku is de ned as in (2.6) with u2Xp being replaced
by u2X. It is then not di cult to see that there is 2(0;1) such that
< inf
x2RN
u+(x) and u(1;x;u0) for jx j r :
Let v0(x) = 1 u(1;x;u0). Then by (6.25),
lim inft!1 inf
jx j ct
(u(t;x;v0) u+(x)) = 0: (6.26)
By (H2) and Proposition 3.1, we have
u(t+ 1;x;u0) u(t;x; v0) u(t;x;v0): (6.27)
By (6.26) and (6.27), there is T > 0 such that
u(T;x;u0) for jx j r : (6.28)
By (6.25) and (6.28),
lim inf
t!1
inf
jx j ct
(u(t+T;x;u0) u+(x)) = 0: (6.29)
(2) then follows from the arbitrariness of c with 0 sup
2SN 1
c ( ).
First, let u0 be as in (1). For every given 2SN 1, there is ~u0( ; )2X+( ) such that
u0( ) ~u0( ; ). Then by Proposition 3.1 and Proposition 3.2,
0 u(t;x;u0;z) u(t;x; ~u0( ; );z)
for t> 0, x2RN, and z2RN. It then follows from Proposition 6.3 that
0 lim sup
x ct;t!1
u(t;x;u0;z) lim sup
x ct;t!1
u(t;x; ~u0( ; );z) = 0
uniformly in z2RN.
Take any c0 >c. Consider all x2RN withkxk= c0. By the compactness of @B(0;c0) =
fx2RNjkxk= c0g, there are 1; 2; ; K 2SN 1 such that for every x2@B(0;c0), there
is k (1 k K) such that x k c. Hence for every x2RN with kxk c0t, there is
1 k K such that x k = kxkc0
c0
kxkx
k kxkc0 c ct. By the above arguments,
0 lim sup
x k ct;t!1
u(t;x;u0;z) lim sup
x k ct;t!1
u(t;x; ~u0( ; k);z) = 0
uniformly for z2RN, k = 1;2; K. This implies that
lim sup
kxk c0t;t!1
u(t;x;u0;z) = 0 uniformly in z2RN:
Since c0 >c and c> sup
2SN 1
c ( ) are arbitrary, we have that for c> sup
2SN 1
c sup( ),
lim sup
kxk ct;t!1
u(t;x;u0;z) = 0 uniformly in z2RN:
(2) First of all, for given x0 2RN and r> 0, let B(x0;r) =fx2RNjkx x0k><
>>:
v0(s 2) for s 0
v0( s 2) for s 0:
For a given B > 0, let
uB0 (x) =
8
>><
>>:
~v0(kxkB ) for kxk B
~v0(1 +kxk B) for kxk>B:
Fix 0 maxf 1c c1; 1c1 c2; 1c2 c3; 1c3 c4g such that for every 2SN 1,
u(t;x;u 0( );z) > for t T ; x c3t; z2RN: (6.30)
In fact, for every 2SN 1, by Proposition 6.2, there is T( ) > 0 such that
u(t;x;u 0( );z) > for t T( ); x ct; z2RN:
In particular,
u(T( );x;u 0( );z) > for x cT( ); z2RN:
84
Let 0 < 0 < (c c1)T( ). Then
u(T( );x;u 0( );z) > for x2cl B(c1T( ) ; 0) ; z2RN:
Note that for given > 0, n2SN 1, and zn2RN with n! and zn!z,
ku n0 (z zn + ) u 0( )kX( ) !0
as n!1. Observe also that
u(T( );x;u n0 ;zn) = u(T( );x;u n0 ;z +zn z)
= u(T( );x+zn z;u n0 (z zn + );z):
Then by Proposition 3.3,
u(T( );x;u n0 ;zn)!u(T( );x;u 0;z)
as n!1 uniformly for x in compact sets. This implies that there is > 0 such that for
2B( ; )\SN 1,
c1T( ) 2B(c1T( ) ; 0)
and
u(T( );x;u 0;z) >
for x2cl B(c1T( ) ; 0) and z2RN. Hence for 2B( ; )\SN 1,
u(T( );c1T( ) ;u 0;z) > for z2RN: (6.31)
85
Observe that
u(T( ); +c1T( ) ;u 0;z) = u(T( );c1T( ) ;u 0( + );z + )
= u(T( );c1T( ) ;u 0( );z + )
for all 2RN with = 0. Then by (6.31), it follows for 2B( ; )\SN 1 that
U(T( );x;u 0;z) > for x = c1T( ); z2RN: (6.32)
Observe also that for each x2RN with x c1T( ), there is x02RN such that x0 0,
(x+x0) = c1T( ), and
u 0( x0) u 0( ):
Then by (6.32), one has for 2B( ; )\SN 1 that
u(T( );x;u 0;z) = u(T( );x+x0;u 0( x0);z x0) > for x c1T( ): (6.33)
Therefore, for 2B( ; )\SN 1,
u(T( );x+c2T( ) ;u 0;z) > for x (c1 c2)T( ):
This implies that
U(T( ); +c2T( ) ;u 0;z) u 0( )
for 2B( ; )\SN 1. It then follows by induction, from Proposition 3.1 and Proposition
3.2 that
u(nT( ); +nc2T( ) ;u 0;z) u 0( )
for n = 1;2; and 2B( ; )\SN 1.
86
By the arguments of Proposition 6.4 (1), there is T ( ) such that for 2B( ; )\SN 1,
u(t;x;u 0;z) > for t T ( ); x c3t; z2RN:
Then by the compactness of SN 1, there is T such that (6.30) holds for every 2SN 1.
This proves the claim.
Now given 2SN 1 and B >c3T , let
uB; 0 (x) = uB0 (x+ (B + 1) );
uB; 0 (x) = uB0 (x c3T );
and
~uB; 0 (x) = minf uB; 0 (x);uB; 0 (x)g:
Then for every > 0,
k~uB; 0 (x) u 0(x)kX( ) !0
as B!1. Hence
u(T ;x; ~uB; 0 ;z)!u(T ;x;u 0;z)
as B!1 uniformly for x in bounded sets and z2RN. By (6.30) and arguments similar
to those in (6.31), there is B+( ) and ~ + > 0 such that for B B+( ) > c3T and ~ 2
B( ;~ + )\SN 1,
u(T ;c3 ~ T ; ~uB;~ 0 ;z) >
for z2RN.
Observe that for every 2[ c3T ;B + 1] and ~ 2B( ;~ + )\SN 1,
uB0 ( + ~ ) ~uB;~ 0 ( )
87
and
u(T ;c3 ~ T + ~ ;uB0 ;z) = u(t;c3 ~ T ;uB0 ( + ~ );z + ~ ):
It then follows that
u(T ; ~ ;uB0 ;z) > for 0 ~ B + 1 +c3T :
Similarly, there is B ( ) > c3T and ~ such that for B > B ( ) and ~ 2B( ;~ )\
SN 1,
u(T ; ~ ;uB0 ;z) > for B 1 c3T ~ 0:
Let B( ) = maxfB+( );B ( )g and ~ = minf~ ;~ + g. Then we have that for every
~ 2B( ;~ )\SN 1,
u(T ; ~ ;uB0 ;z) > for B 1 c3T B + 1 +c3T :
By the compactness of SN 1, there is B >c3T such that
u(T ;x;uB0 ;z) > for kxk B + 1 +c3T ; z2RN:
Hence for B B ,
u(T ; ;uB0 ;z) uB+c3T 0 ( ) for z2RN:
By induction, Proposition 3.1 and Proposition 3.2, one obtains for B B that
u(nT ; ;uB0 ;z) uB+c3nT 0 ( ) for z2RN:
This together with Proposition 6.4 (3) implies that for every B B ,
lim inf
kxk c4t;t!1
(u(t;x;uB0 ;z) u+(x+z)) = 0
88
uniformly in z2RN.
Finally, we can prove that r can be chosen to be independent of by arguments similar
to those in Theorem E(2). This completes the proof.
6.3 Spreading Speeds in the General Case
In this section, we investigate the existence and characterization of the spreading speeds
of (1.2) without the assumption (H4).
Lemma 6.1. Assume (H1) - (H3). For every 2SN 1, there is ( )2(0;1) such that
0( ; ( );a0)
( ) = inf >0
0( ; ;a0)
:
Proof. First, it is not di cult to see that 0( ; ;a0) is continuous in . By (H2), 0( ;0;a0) >
0 and hence lim !0+ 0( ; ;a0) =1. By Theorem 4.1 and Theorem B, lim !1 0( ; ;a0) =1.
The lemma then follows.
Proof of Theorem D. (1) First, we prove that c sup( ) inf >0 0( ; ;a0) .
Let an( )2CN(RN)\Xp be such that an satis es (H5),
an a0 for n 1 and kan akXp !0 as n!1:
Then by Lemma 5.1,
0( ; ;an)! 0( ; ;a0) as n!1:
Let n be the positive eigenfunction of K ; I + an( )I corresponding to ( ; ;an) =
0( ; ;an) with k nkXp = 1. Note that
uf(x;u) uf(x;0) an(x)u for x2RN; u 0
89
and
(t; ;0;an)u
;
(x) = e (x 0( ; ;an)
t) n(x);
where u ; (x) = e x n(x). Hence by Proposition 3.1 and Proposition 3.2, for any > 0,
u(t;x;u ; ) e (x 0( ; ;a
n)
t) n(x) for t 0:
This implies that
c sup( ) 0( ; ;a
n)
8 > 0; n 1
and then
c sup( ) 0( ; ;a0) 8 > 0:
Therefore,
c sup( ) inf >0 0( ; ;a0) : (6.34)
Next, we prove c inf( ) inf >0 0( ; ;a0) . For any > 0, there is 0 > 0 such that such
that
f(x;u) f(x;0) for x2RN; 0 ____0 ( ; ;an) :
Applying the arguments in Theorem D, there is u0( ;z)2X+( ) such that
lim inf
x ! 1
inf
z2RN
u0(x;z) > 0
90
and
u(1;x;u0( ;z);z) u0(x (c n( ) ) ; (c n( ) ) +z) 8z2RN:
This implies that
u(m;x;u0( ;z);z) u0(x m(c n( ) ) ;m(c n( ) ) +z) 8m 1; z2RN:
Then by Proposition 6.4,
c inf( ) c n( ) :
By Lemma 5.1,
c inf( ) inf >0 0( ; ;a0) 2 :
Letting !0, by Lemma 6.1, we have
c inf( ) inf >0 0( ; ;a0) : (6.35)
By (6.34) and (6.35),
c sup( ) = c inf( ) = inf >0 0( ; ;a0) :
Hence c ( ) exists and
c ( ) = inf >0 0( ; ;a0) :
(2) (3) (4) They can be proved by arguments similar to those in last section.
Proof of Theorem E. (1) Fix c> maxfc ( );c ( )g. As in the proof of Theorem D (1), let
an( )2CN(RN)\Xp be such that an satis es (H5),
an a0 for n 1 and kan akXp !0 as n!1:
91
Choose > 0 and n 1 such that
0( ; ;an)
1 such that
u0(x) Me x n(x);
where n(x) is the positive eigenfunction of K ; I +an( )I corresponding to ( ; ;an) =
0( ; ;an) with k nkXp = 1. By arguments similar to those in Theorem E (1),
u(t;x;u0) e (x 0( ; ;a
n)
t) n(x) for t 0:
This implies that
lim sup
t!1
sup
x ct
u(t;x;u0) = 0: (6.36)
Similarly, it can be proved that
lim sup
t!1
sup
x ct
u(t;x;u0) = 0: (6.37)
(1) thus follows from (6.36) and (6.37).
(2) It follows from arguments in last section.
Proof of Theorem F. (1) It can be proved by arguments similar to those in last section.
(2) It can be proved by arguments similar to those in last section.
6.4 E ects of Spatial Variations on Spreading Speeds
In this section, we will investigate the e ects of spatial variations on spreading speeds
and prove the Theorem G. Let
f(u) = 1
jDj
Z
D
f(x;u)dx:
92
We assume that f satisfy f(0) > 0 .
Then f is also monostable type functions. Let c ( ; f) be the spreading speeds of the
following averaged equations of (6.38),
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +u f(u): (6.38)
Proof of Theorem G. First, let a0(x) = f(x;0), and by Theorem D, for any 2SN 1, there
is ( ) > 0 such that
c ( ;f) = 0(
( ); ;a0(x))
( ) :
Then by Theorem B, Lemma 5.2 and Theorem D,
c ( ;f) = 0(
( ); ;a0(x))
( )
( ( ); ; a)
( ) c
( ; f):
Now assuming (H4), for some 2SN 1, c ( ;f) = c ( ; f), then we must have
( ( ); ;a0(x))
( ) =
( ( ); ; a)
( ) :
By Theorem B again, we must have a0(x) a.
Theorem G shows that it is a generic scenario that spatial variation increases the spread-
ing speed.
93
Chapter 7
Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations
In this chapter, we explore the traveling wave solutions of (1.2) and prove Theorems
H-J. To this end, we rst construct some sub- and super-solutions to be used in the proofs
of the main results. We then study the existence, uniqueness, and stability of traveling wave
solutions of (1.2). The results of this chapter have been submitted for publication (see [58]).
Throughout this chapter, we assume that (H1)-(H4). Biologically, we are only interested
in nonnegative solutions of (1.2). Hence, without loss of generality, we make the following
technical assumption throughout this chapter, f(x;u) = f(x;0) for u 0.
7.1 Sub- and Super-solutions
Let a0(x) = f(x;0).
For given 2SN 1, let ( ) be such that
c ( ) = 0( ;
( );a0)
( ) :
Fix 2SN 1 and c>c ( ). Let 0 < < 1 < minf2 ; ( )gbe such that c = 0( ; ;a0) and
0( ; ;a0)
>
0( ; 1;a0)
1 >c
( ): Let ( ) and 1( ) be positive eigenfunctions ofK ; I+a0( )I
associated to 0( ; ;a0) and 0( ; 1;a0) withk ( )kXp = 1 andk 1( )kXp = 1, respectively.
If no confusion occurs, we may write 0( ; ;a0) as ( ).
For given d1 > 0, let
v1(t;x;z;T;d1) = e (x +cT ct) (x+z) d1e 1(x +cT ct) 1(x+z): (7.1)
We may write v1(t;x;z;T) for v1(t;x;z;T;d1) for xed d1 > 0 or if no confusion occurs.
94
Proposition 7.1. For any z 2 RN and T > 0, v1(t;x;z;T) is a sub-solution of (2.15)
provided that d1 is su ciently large.
Proof. First of all, let ? = e (x +cT ct) (x + z) and ?1 = d1e 1(x +cT ct) 1(x + z). Let
M = max
x2RN
(x)(> 0). Let L> 0 be such that fu(x + z;u) L for 0 u M. Let d0 be
de ned by
d0 = maxf
max
x2RN
(x)
min
x2RN
1(x);
Lmax
x2RN
2(x)
( 1c ( 1)) min
x2RN
1(x)g
Fix z 2 RN and T > 0. We prove that v1(t;x;z;T) is a sub-solution of (2.15) for
d1 d0, that is, for any (t;x)2R RN,
@v1
@t [
Z
RN
k(y x)v1(t;y;z;T)dy v1(t;x;z;T) +f(x+z;v1(t;x;z;T))v1(t;x;z;T)] 0:
(7.2)
First, for (t;x) 2R RN with v1(t;x;z;T) 0, f(x + z;v1(t;x;z;T)) = f(x + z;0).
Hence
@v1
@t [
Z
RN
k(y x)v1(t;y;z;T)dy v1(t;x;z;T) +f(x+z;v1(t;x;z;T))v1(t;x;z;T)]
= ( 1c ( 1))?1 0:
Therefore (7.2) holds for (t;x)2R RN with v1(t;x;z;T) 0.
Next, consider (t;x) 2 R RN with v1(t;x;z;T) > 0. By d1 d0, we must have
x + cT ct 0. Then v1(t;x;z;T) e (x +cT ct) (x + z) (x + z) M. Note that
for 0 0,
@v1
@t [
Z
RN
k(y x)v1(t;y;z;T)dy v1(t;x;z;T) +f(x+z;v1)v1(t;x;z;T)]
= c? 1c?1 [
Z
RN
k(y x)v1(t;y;z;T)dy v1(t;x;z;T) +f(x+z;v1)v1(t;x;z;T)]
=( c ( ))? ( 1c ( 1))?1 +f(x+z;0)v1(t;x;z;T) f(x+z;v1)v1(t;x;z;T)
= ( 1c ( 1))?1 fu(x+z;y)(? ?1)2 for some y2(0;M)
( 1c ( 1))?1 fu(x+z;y)(?)2
=[ ( 1c ( 1)) fu(x+z;y)(?)
2
?1 ]?1
0:
Hence (7.2) also holds for (t;x) 2 R RN with v1(t;x;z;T) > 0. The proposition then
follows.
Let (0) be the principal eigenvalue and 0 be the positive principal eigenfunction
of K I + a0( )I with k 0kXp = 1. Observe that there exists su ciently large M > 0
such that v1(t;x0;z;T) 12e (x +cT ct) (x + z) for x + cT ct > M. Thus we have
v1(t;x0;z;T) 12e (x +cT ct) min
x2RN
f (x)g for x + cT ct > M. For any 0 > 0, let
M1 be such that M1 M > 0 and ^b = 12e M1 minx (x). Then we have v1(t;x0;z;T)
1
2e
(x +cT ct) min
x2RN
f (x)g ^b for any M b 0(x+z) for any M 0
k(z)dz ^ := 12 (0) min
x2RN
f 0(x)g=max
x2RN
f 0(x)g.
Let 0 < b 1 and M1 > M > 0 be such that v1(t;x0;z;T) > b 0(x + z) for any
M 0 and (7.3) holds. Let
u(t;x;z;T;d1;b) =
8
>><
>>:
maxfb 0(x+z);v1(t;x;z;T;d1)g for x +cT ct M1g. Note that D0 fxjx + cT ct Mg.
If x 2 D0 and y 2 D2, then ky xk (y x) M1 M 0, which implies
fyjy2D2;x2D0g fyjky xk> 0g. Thus,
Z
D2
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy
Z
ky xk> 0
k(y x)b 0(y +z)dy
Z
kzk> 0
k(z) max
x2RN
fb 0(x)gdy
^ max
x2RN
fb 0(x)g
=12 (0)b min
x2RN
f 0(x)g
Thus, for x2D0,
Z
D2
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy 12 (0)b 0(y +z): (7.6)
If x2D0, we have
98
@u(t;x;z;T;d1;b)
@t [
Z
RN
k(y x)u(t;y;z;T;d1;b)dy u(t;x;z;T;d1;b)
+f(x+z;u(t;x;z;T;d1;b))u(t;x;z;T;d1;b)]
= [
Z
RN
k(y x)u(t;y;z;T;d1;b)dy b 0(x+z) +f(x+z;b 0(x+z))b 0(x+z)]
= [
Z
RN
k(y x)b 0(y +z)dy b 0(x+z) +f(x+z;0)b 0(x+z)]
+
Z
RN
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy
+ [f(x+z;0)b 0(x+z) f(x+z;b 0(x+z))b 0(x+z)]
= (0)b 0(x+z) +
Z
RN
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy f2(x+z; )(b 0(x+z))2
= (0)b 0(x+z) +
Z
D1
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy
+
Z
D2
k(y x)[b 0(y +z) u(t;y;z;T;d1;b)]dy f2(x+z; )(b 0(x+z))2:
Together with the inequalities (7.3), (7.5) and (7.6), we have for x2D0,
@u(t;x;z;T)
@t
Z
RN
k(y x)u(t;y;z;T)dy u(t;x;z;T) +u(t;x;z;T)f(x+z;u(t;x;z;T)):
(7.7)
By (7.4) and (7.7), we proved the claim.
Therefore, u(t;x;z;T) is a sub-solution of (2.15).
For given d2 0, let
v(t;x;z;T;d2) = e (x +cT ct) (x+z) +d2e 1(x +cT ct) 1(x+z)
and
u(t;x;z;T;d2) = minf v(t;x;z;T;d2);u+(x+z)g:
We may write v(t;x;z;T) and u(t;x;z;T) for v(t;x;z;T;d2) and u(t;x;z;T;d2), respectively,
if no confusion occurs.
Proposition 7.3. For any d2 0, z2RN, and T > 0, u(t;x;z;T) is a super-solution of
(2.15).
99
Proof. It su ces to prove that v(t;x;z;T) is a super-solution.
Let ?2 = d2e 1(x +cT ct) 1(x+z). By direct calculation, we have
@ v
@t [
Z
RN
k(y x) v(t;y;z;T)dy v(t;x;z;T) +f(x+z; v) v(t;x;z;T)]
@ v@t [
Z
RN
k(y x) v(t;y;z;T)dy v(t;x;z;T) +f(x+z;0) v(t;x;z;T)]
=( 1c ( 1))?2
0:
The proposition thus follows.
In the rest of this section, we x d 1 1, d 2 0, and 0 __**t1 > 0,
u(t2 +t;x;u 0;z;t2;z) u(t1 +t;x;u 0;z;t1;z) 8t> t1; x2RN;
(2) u(t2 +t;x;u+0;z;t2;z) u(t1 +t;x;u+0;z;t1;z) 8t> t1; x2RN:
Proof. (1) For given z2RN and t2 >t1 > 0, by Proposition 7.2,
100
u(t2 t1;x;u 0;z;t2;z) u(t2 t1;x;z;t2)
= u 0;z;t2 (t2 t1)(x)
= u 0;z;t1(x):
Hence
u(t2 +t;x;u 0;z;t2;z) = u(t1 +t;x;u(t2 t1; ;u 0;z;t2;z);z)
u(t1 +t;x;u 0;z;t1;z):
(1) is thus proved.
(2) It follows by arguments similar to those in (1) and Proposition 7.3.
7.2 Existence and Uniqueness of Traveling Wave Solutions
In this section, we investigate the existence of traveling wave solutions of (1.2) and prove
Theorem H.
Let u 0;z;T be as in (7.8). Let
(x;z) = lim !1u( ;x;u 0;z; ;z) (7.9)
and
U (t;x;z) = lim !1u(t+ ;x;u 0;z; ;z): (7.10)
By Proposition 7.4, the limits in the above exist for all t2R and x;z2RN. Moreover, it is
easy to see that (x;z) is lower semi-continuous in (x;z)2RN RN and +(x;z) is upper
semi-continuous.
We will show that u = U+(t;x; 0) and u = U (t;x; 0) are traveling wave solutions of
(1.2) in the direction of with speed c generated by +( ; ) and ( ; ), respectively, and
that ( ; ) := +( ; ) satis es Theorem H(1)-(2).
To this end, we rst prove some lemmas.
101
Lemma 7.1. For each z2RN, u(t;x) = U (t;x;z) are entire solutions of (2.15).
Proof. We prove the case that u(t;x) = U+(t;x;z). The other case can be proved similarly.
Fix z2RN. Observe that for any x2RN,
u(t+ ;x;u+0;z; ;z) = u( ;x;u+0;z; ;z) +
Z t
0
Z
RN
k(y x)u(s+ ;y;u+0;z; ;z)dyds
+
Z t
0
u(s+ ;x;u+
0;z; ;z) +u(s+ ;x;u
+
0;z; ;z)f(x+z;u(s+ ;x;u
+
0;z; ;z))
ds
Letting !1, we have
u(t;x) = u(0;x) +
Z t
0
Z
RN
k(y x)u(s;y)dy u(s;x) +u(s;x)f(x+z;u(s;x)) ds:
This implies that u(t;x) is di erentiable in t and satis es (2.15) for all t2R.
Observe that
U (t;x;z) = u(t;x; ( ;z);z) 8t2R; x;z2RN:
Lemma 7.2. u(t;x; ( ;z);z) = (x ct ;z + ct ), lim
x ! 1
( (x;z) u+(x + z)) = 0
and lim
x !1
(x;z)
e x (x+z) = 1 uniformly in z2R
N.
Proof. We prove the lemma for +( ; ). It can be proved similarly for ( ; ).
First of all, we have
u(t;x; +( ;z);z) = lim
!1
u(t;x;u( ;x;u+0;z; ;z);z)
= lim
!1
u(t+ ;x;u+0;z; ;z)
= lim !1u(t+ ;x ct ;u+0;z+ct ;t+ ;z +ct )
= +(x ct ;z +ct ):
Note that
102
e (x ct) (x+z) d1e 1(x ct) 1(x+z) u(t+T;x;z;T)
u(t;x; +( ;z);z)
u(t+T;x;z;T)
= e (x ct) (x+z) +d2e 1(x ct) 1(x+z)
for t 2 R and x;z 2 RN. Thus lim
x ct!1
+(x ct ;z +ct )
e (x ct) (x+z) = 1, which is equivalent to
lim
x !1
+(x;z)
e x (x+z) = 1, uniformly in z2R
N.
We now prove that lim
x ! 1
+(x;z) u+(x + z) = 0 uniformly in z2RN. Observe
that there is M > 0 such that
U+(t;x;z) U (t;x;z) b 0(x+z) for x ct M; z2RN:
By Proposition 3.3, for any > 0, there are T > 0 and 2R such that
jU+(T;x;z) u+(x+z)j< for x ; z2RN:
This implies that
j +(x;z) u+(x+z)j for x +cT; z2RN
and hence lim
x ! 1
+(x;z) u+(x+z) = 0 uniformly in z2RN.
Corollary 7.1. Both +( ; ) and ( ; ) generate traveling wave solutions of (1.2) in the
direction of with speed c.
Proof. First of all, by Lemmas 7.1 and 7.2, both +( ; ) and ( ; ) satisfy (2.17) and (2.18).
Next, for any x;x02RN with x = x0 , z2RN, and 2R, we have
u( ;x0;u 0;z x0; ( );z x0) = u( ;x;u 0;z x0; ( +x0 x);z x0 + (x0 x))
= u( ;x;u 0;z x; ( );z x):
103
This implies that ( ; ) satis es (2.19).
Observe now that u 0;z+piei; = u 0;z; for any 2R and z 2RN. It then follows that
(x;z +piei) = (x;z) and hence ( ; ) satis es (2.20).
Therefore, both +( ; ) and ( ; ) generate traveling wave solutions of (1.2) in the
direction of with speed c.
Lemma 7.3. lim
x ct! 1
U t (t;x;z) = 0 uniformly in z2RN.
Proof. Note that
U t (t;x;z) =
Z
RN
k(y x)U (t;y;z)dy U (t;x;z) +U (t;x;z)f(x+z;U (t;x;z))
and thus
lim
x ct! 1
U t (t;x;z) = lim
x ct! 1
h
U t (t;x;z)
Z
RN
k(y)u+(y +x+z)dy +u+(x+z)
u+(x+z)f(x+z;u+(x+z))
i
= lim
x ct! 1
hZ
RN
k(y) U (t;x+y;z) u+(x+y +z) dy
U (t;x;z) u+(x+z)
+ U (t;x;z)f(x+z;U (t;x;z)) u (x+z)f(x+z;u+(x+z))
i
It su ces to prove that lim
x ct! 1
Z
RN
k(y) U (t;x+y;z) u+(x+y+z) dy!0 uniformly in
z2RN. For any > 0. Since U (t;x+y;z) u+(x+y+z) is bounded and k( ) satis es (H1),
then there exists a ^ > 0 such that
Z
kyk>^
k(y) U (t;x+y;z) u+(x+y+z) dy< 2 for z2
RN. Since lim
x ct! 1
[U (t;x;z) u+(x + z)] = 0 uniformly in z 2 RN, there exists an
L > 0 such that U (t;x;z) u+(x + z) < 2 for x ct < L and z2RN. Thus,Z
kyk ^
k(y) U (t;x+y;z) u+(x+y+z) dy< 2 for x ct< L ^ and z2RN.
Therefore,
Z
RN
k(y) U (t;x+y;z) u+(x+y+z) dy< for x ct< L ^ and z2
RN. This completes the proof.
104
Lemma 7.4. lim
x ct!1
U t (t;x;z)
e (x ct) (x+z) = c uniformly in z2R
N.
Proof. We prove the lemma for U+(t;x;z). It can be proved similarly for U (t;x;z).
First, let U(t;x;z) = U+(t;x;z). By Lemma 7.2, for any > 0, there is M > 0 such
that for any x;z2RN and t2R with x ct M,
U(t;x;z)
e (x ct) (x+z)
< (7.11)
and
jf(x+z;U(t;x;z)) f(x+z;0)j< : (7.12)
Observe that
c (x+z) =
Z
RN
e (y x) k(y x) (y +z)dy (x+z) +a0(x+z) (x+z) (7.13)
for all x;z2RN, where a0(x+z) = f(x+z;0), and
Ut(t;x;z) =
Z
RN
k(y x)U(t;y;z)dy U(t;x;z) +U(t;x;z)f(x+z;U(t;x;z)) (7.14)
for all t2R and x;z2RN. By (7.11)-(7.14), we have
Ut(t;x;z)
e (x ct) (x+z) c
= 1
(x+z)
Z
RN
e (y x) k(y x) U(t;y;z)e (y ct) (y +z) dy
U(t;x;z)e (x ct) (x+z)
+ U(t;x;z)e (x ct) (x+z) f(x+z;U(t;x;z))
+ (x+z) f(x+z;U(t;x;z)) f(x+z;0)
Z
RN
e (y x) k(y x)dy
+ 1 +jf(x+z;U(t;x;z))j+ (x+z)
105
for all x;z 2 RN and t2 R with x ct M + 0, where 0 is the nonlocal dispersal
distance in (1.2). It then follows that
lim
x ct!1
U t (t;x;z)
e (x ct) (x+z) = c
uniformly in z2RN.
Proof of Theorem H. Let (x;z) = +(x;z) andU(t;x;z) = U+(t;x;z). Note thatU(t;x;z) =
u(t;x; ( ;z);z)). We show that ( ; ) and U( ; ; ) satisfy Theorem H(1) and (2), respec-
tively.
(1) It follows from Corollary 7.1 and Lemma 7.2.
(2) By Lemmas 7.3 and 7.4, we only need to prove that Ut(t;x;z) > 0 for all (t;x;z)2
R RN RN.
For any t1 0g has positive Lebesgue measure. Note that
v(t;x;z) satis es
vt(t;x;z) =
Z
R
k(y x)v(t;y;z)dy v(t;x;z) +a(t;x;z)v(t;x;z) (7.15)
106
where a(t;x;z) = f(x+z;u(t;x; +( ;z);z)) +u(t;x; +( ;z);z)fu(x+z;u(t;x; +( ;z);z)).
Then by Proposition 3.1, we have
v(t;x;z) > 0 8t2R; x;z2RN:
This implies that Ut(t;x;z) > 0 for all t2R and x;z2RN.
Next, we investigate the uniqueness and continuity of traveling wave solutions of (1.2)
and prove Theorem I by the \squeezing" techniques developed in [9] and [25].
Throughout this section, we x 2SN 1 and c>c ( ). Let be such that
c ( ) = 0( ;
;a0)
<
0(~ ; ;a0)
~ 8~ 2(0;
):
We x c>c ( ) and 2(0; ) with 0( ; ;a0) = c and assume that U (t;x;z) and (x;z)
are as in section 7.2. We put (x;z) = +(x;z) and U(t;x;z) = U+(t;x;z). Let U1(t;x;z) =
u(t;x; 1( ;z);z)( 1(x ct ;z +ct )).
We rst prove some lemmas, some of which will also be used in next section. By Lemmas
7.2 and 7.4, there is M0 > 0 such that
0 < sup
x ct M0;z2RN
U(t;x;z)
Ut(t;x;z) <1: (7.16)
Observe that there is 0 > 0 such that
U(t;x;z) 0 for x ct M0: (7.17)
Let
0 = inf
0**__ 0 such that for each
2(0; 0),
107
H (t;x;z) = (1 e t)U(t l e t;x;z);8t 0 x;z2RN
are super-/sub-solution of (2.15).
Proof. First we prove that H+(t;x;z) is a super-solution of (2.15). Let h = e t and
= t l e t. Then
H+(t;x;z) = (1 +h)U( ;x;z);8t 0; x;z2RN:
By direct calculation, we have
@H+(t;x;z)
@t [
Z
RN
k(y x)H+(t;y;z)dy H+(t;x;z) +H+(t;x;z)f(x+z;H+(t;x;z))]
= hU( ;x;z) + (1 +l h)[(K I)H+ +f(x+z;U)H+] [(K I)H+ +f(x+z;H)H+]
= hU( ;x;z) +l h[(K I)H+ +f(x+z;U)H+] + [f(x+z;U) f(x+z;H)]H+
= hU( ;x;z) +l h(1 +h)Ut( ;x;z) + [f(x+z;U) f(x+z;H+)](1 +h)U( ;x;z)
= h U( ;x;z)[ 1 +l(1 +h)Ut( ;x+z)U( ;x+z) fu(x+z;u ( ;x;z))(1 +h)U( ;x;z)= ];
where u ( ;x;z) is some number between U( ;x;z) and H+(t;x;z). We only need to prove
that
1 +l(1 +h)U ( ;x;z)U( ;x;z) fu(x+z;U ( ;x;z))(1 +h)U( ;x)= 0 (7.19)
for all t 0 and x;z2RN.
If t 0 and x2RN are such that x c M0, by (7.17), (7.18), and the fact that
Ut( ;x;;z) > 0, (7.19) holds.
If t 0 and x2RN are such that x c M0, and l sup
x c M0
U( ;x;z)
Ut( ;x;z), then
(7.19) also holds.
By the similar arguments above, we can prove that H (t;x;z) is a sub-solution of (2.15).
This completes the proof.
108
Lemma 7.6. Let 0 2(0;1) and 2(0;(1 0) 0) be given and l be as in Lemma 7.5. For
any given 0 < 1 0, there exists constant M1( 1) > 0 such that for all 2(0; 1]
(1 )U(t+3l ;x;z) U(t;x;z) (1+ )U(t 3l ;x;z) 8t2R; x;z2RN; x ct M1( 1):
Proof. Let h(s) = (1+s)U(t 3ls;x;z). Then, h0(s) = U(t 3ls;x;z) 3lUt(t 3ls;x;z). By
Lemma 7.3, there exists a M( 1) > 0 such that h0(s) > 0 for s2[ 1; 1], x ct M1( 1),
and z2RN. Hence, the lemma follows.
Lemma 7.7. For any > 0, there exists a constant C( ) 1 such that
U1(t 2 ;x;z) U(t;x;z) U1(t+ 2 ;x;z) 8t2R; x;z2RN; x ct C( ):
Proof. It follows from the fact that
lim
x ct!1
U1(t;x;z)
e (x ct) (x+z) = limx ct!1
U1(t;x;z)
U(t;x;z)
U(t;x;z)
e (x ct) (x+z)
= lim
x ct!1
U(t;x;z)
e (x ct) (x+z)
= 1
uniformly in z2RN.
Lemma 7.8. Let 0 2(0;1) and 0, l be as in Lemma 7.5. For any given 2(0; 0), there
is > 0 such that
(1 e t)U(t +l e t;x) U1(t;x;z) (1 + e t)U(t+ l e t;x;z)
for all x;z2RN and t 0.
Proof. First by Theorem C(1) and 3.1,
0 ____ 0;t1 > 0; and M 2R be given. Suppose that W (t;x;t1;z) are the
solution of (2.15) with initial
W (0;x;t1;z) = U(t1 ;x;z)&(x ct1 M) +U(t1 2 ;x;z)(1 &(x ct1 M));
where &(s) = 0 for s 0 and &(s) = 1 for s> 0. Then
W+(1;x;t1;z) (1 + )U(t1 + 1 + 2 3l ;x;z)
and
W (1;x;t1;z) (1 )U(t1 + 1 2 + 3l ;x;z)
for all x;z2RN with x c(1 +t1) M provided that 0 < 1.
110
Proof. We give a proof for W (1;x;t1;z). The case of W+ can be proved similarly. Note
that
W (0;x;t1;z) U(t1 2 ;x;z) 8x;z2RN:
It then follows that
W (1;x;t1;z) >U(1 +t1 2 ;x;z) 8x;z2RN:
Take an 1 2(0; 0]. By Lemma 7.6, for any 2(0; 1],
W (1;x;t1;z) > (1 )U(1 +t1 2 + 3l ;x;z) 8x c(t1 + 1) M( 1); z2RN:
We claim that for 0 < 1,
W (1;x;t1;z) > (1 )U(1 +t1 2 + 3l ;x;z) 8x c(t1 + 1)2[ M( 1);M]; z2RN:
In fact, let W(t;x;z) = W (t;x;t1;z) U(t+t1 2 ;x;z) and
h = inf
t2[0;1];x;z2RN
f[W (t;x;t1;z)f(x+z;u(t;x;u0;z;z))
U(t+t1 2 ;x;z)f(x+z;U(t+t1 2 ;x;z))]
1W (t;x;t
1;z) U(t+t1 2 ;x;z)
g:
Then
W(0;x;z) =
8
>><
>>:
U(t1 ;x;z) U(t1 2 ;x;z) for x ct1 >M
0 for x ct1 M
and
Wt(t;x;z)
Z
RN
k(y x)W(t;y;z)dy W(t;x;z) +hW(t;x;z) 8t2[0;1]; x;z2RN:
It then follows that
W(1; ;z) e 1+h(W(0; ;z) +KW(0; ;z) + K
2
2! W(0; ;z) + );
111
where KW(0; ;z) is de ned as in (2.6) with u being replaced by W(0; ;z). By Lemma 7.2,
there are ~ > 0 and ~M > 0 such that
U(t1 ;x;z) U(t1 2 ;x;z) ~ 8x;z2RN with ~M x ct1 ~M + 1: (7.20)
This implies that
W(1;x;z) U(1 +t1 2 + 3l ;x;z) U(1 +t1 2 ;x;z) (7.21)
for x c(t1 + 1) 2 [ M( 1);M] and z 2RN provided that 0 < 1. By (7.20) and
(7.21), we have
W (1;x;t1;z) = W(1;x;z) +U(1 +t1 2 ;x;z)
U(1 +t1 2 + 3l ;x;z)
(1 )U(1 +t1 2 + 3l ;x;z)
for x c(1 +t1) M and z2RN provided that 0 < 1.
Proof of Theorem I. (1) Let
A+ =f 0j lim sup
t!1
sup
x;z2RN
U1(t;x;z)
U(t+ 2 ;x;z) 1g
and
A =f 0j lim inft!1 inf
x;z2RN
U1(t;x;z)
U(t 2 ;x;z) 1g:
By Lemma 7.8, A 6=;. Let
+ = inff j 2A+g; = inff j 2A g:
We rst claim that 2A . In fact, let n2A+ be such that n! +. Then for any
0 < < 1, there are tn!1 such that
U1(t;x;z)
U(t+ 2 n;x;z) 1 + 8x;z2R
N; t tn
112
and
U(t+ 2 +;x;z) U(t+ 2 n;x;z)
U(t+ 2 n;x;z) > 8n 1; t2R; x;z2R
N:
Observe that
U1(t;x;z)
U(t+ 2 +;x;z) =
U1(t;x;z)
U(t+ 2 n;x;z)
U(t+ 2 n;x;z)
U(t+ 2 +;x;z)
and
U(t+ 2 n;x;z)
U(t+ 2 +;x;z) =
1
1 + U(t+2 +;x;z) U(t+2 n;x;z)U(t+2 n;x;z)
11
1 + 8n 1:
Fix n 1. Then
sup
x;z2RN
U1(t;x;z)
U(t+ 2 +;x;z) (1 + )
2 8t tn:
This implies that + 2A+. Similarly, we have 2A .
Next we claim that = 0. Assume that > 0. Note that
lim inft!1 inf
x;z2RN
U1(t;x;z)
U(t 2 ;x;z) 1:
Hence for any > 0, there is t0 > 0 such that
U1(t0;x;z)
U(t0 2 ;x;z) 1 8x;z2R
N:
This implies that
U1(t0;x;z) (1 )U(t0 2 ;x;z) U+(t0 2 ;x;z) ^
where ^ = maxt;x;zU+(t;x;z). By Lemma 7.7, for x ct0 M := C( =2),
U1(t0;x;z) U(t0 ;x;z):
113
This implies that
U1(t0;x;z) U(t0 2 ;x;z)(1 (x ct0 M)) +U(t0 ;x;z) (x ct0 M) ^ :
Note that there is K > 0 such that U1(t;x;z)+^ eKt is a super-solution of (2.15) for t2[0;1]
provided that 0 < ^ 1. Then by Lemma 7.9, for 0 < 1 and 0 < 1,
U1(t0 + 1;x;z) + ^ eK (1 )U(t0 + 1 2 + 3l ;x;z) 8x c(t0 + 1) M; z2RN;
where l is as in Lemma 7.5. Hence for 0 < 1,
U1(t0 + 1;x;z) (1 2 )U(t0 + 1 2z + 3l ;x;z) 8x c(t0 + 1) M; z2RN:
By Lemma 7.7 again, for x c(t0 + 1) M , z2RN, and 0 < 1,
U1(t0 + 1;x;z) >U(t0 + 1 ;x;z)
(1 2 )U(t0 + 1 ;x;z)
(1 2 )U(t0 + 1 2 + 3l ;x;z):
Therefore for 0 < 1,
U1(t0 + 1;x;z) (1 2 )U(t0 + 1 2 + 3l ;x;z) 8x;z2RN:
By Lemma 7.5,
U1(t0 +t+ 1;x;z) (1 2 e t)U(t0 + 1 +t 2 + 2l e t +l ;x;z) 8t 0; x;z2RN:
It then follows that
l 2 2A :
this is a contradiction. Therefore = 0. Similarly, we have + = 0.
114
We now prove that 1(x;z) = (x;z). Recall that U1(t;x;z) = 1(x ct ;z+ct ) and
U(t;x;z) = (x ct ;z +ct ). Hence
inf
x;z2RN
U1(t;x;z)
U(t;x;z) = infx;z2RN
1(x ct ;z +ct )
(x ct ;z +ct )
= inf
x;z2RN
1(x;z)
(x;z)
and
sup
x;z2RN
U1(t;x;z)
U(t;x;z) = supx;z2RN
1(x ct ;z +ct )
(x ct ;z +ct )
= sup
x;z2RN
1(x;z)
(x;z)
This together with = 0 implies that
inf
x;z2RN
1(x;z)
(x;z) = supx;z2RN
1(x;z)
(x;z) = 1:
We then must have 1(x;z) (x;z).
(2) Let 1(x;z) = (x;z)(= U (0;x;z)). By (1), (x;z) = (x;z). Recall that
(x;z) is lower semi-continuous and +(x;z) is upper semi-continuous. We then must
have that (x;z) is continuous in (x;z)2RN RN.
Corollary 7.2. Let (x;z) be as in Theorem H. Then
lim !1u( ;x;u(0; ;z; ;d1;b);z) = lim !1u( ;x; u(0; ;z; ;d2);z) = (x;z)
for all d1 1, d2 > 0, 0 __**c ( ). Let be such that
c ( ) = 0(
; ;a0)
<
0(~ ; ;a0)
~ 8~ 2(0;
):
We x c > c ( ) and 2 (0; ) with 0( ; ;a0) = c. Let U(t;x;z) = U+(t;x;z), where
U+(t;x;z) is as in section 7.2. Put u(t;x) = u(t;x;u0;0), where u0 is as in Theorem I, and
put U(t;x) = U+(t;x; 0). First we prove some lemmas, which are analogues of Lemmas
7.7-7.9.
Lemma 7.10. For any > 0, there exists a constant C0( ) 1 such that
u(t 2 ;x) U(t;x) u(t+ 2 ;x) 8x ct C0( ); t 2 :
Proof. This is an analogue of Lemma 7.7 and can be proved by properly modifying the
arguments in Lemma 7.7. For clarity, we provide a proof in the following.
First we prove that there exists a constant C1( ) 1 such that U(t;x) u(t+ 2 ;x) for
all x ct C1( ). Note that for given > 0, there exists a L> 0 such that
e (x +c ) (x) L:
Choose d1 large enough such that v1 (see (7.1)) is a sub-solution of (1.2) and e (x +c ) (x)
d1e 1(x +c ) 1(x) 0 for all x2RN with x L: Then by Proposition 3.1,
116
u(t;x) e (x c(t )) (x) d1e 1(x c(t )) 1(x) 8x2RN; t 0:
Thus,
u(t+ ;x) v1(t;x) = e (x ct) (x) d1e 1(x ct) 1(x) 8x2RN; t 0:
Observe that there exists a constant C1( ) 1 such that
v1(t+ ;x) U(t;x) 8x ct C1( ):
Therefore,
u(t+ 2 ;x) v1(t+ ;x) U(t;x) 8x ct C1( );t 0:
Next we prove that there exists a constant C2( ) 1 such that U(t;x) u(t 2 ;x)
for all x ct C2( ). Note that there are d2 > 0 and L> 0 such that
u0(x) minfe x (x) +d2e 1x 1(x);Lu+(x)g 8x2RN:
By Proposition 3.1,
u(t;x) minfe (x ct) (x) +d2e 1(x ct) 1(x);u(t;x;Lu+( );0)g 8x2RN;t 0:
On the other hand, we have
lim
x ct!1
U(t;x)
e (x c(t 2 )) (x) = e
2 c > 1
Therefore, there exists C2( ) 1 such that
e (x c(t 2 )) (x) **__ 0
and > 0 such that
(1 e (t t ))U(t +l e (t t );x) u(t;x) (1 + e (t t+))U(t+ + l e (t t+);x)
for all x2RN and t maxft ;t+g.
Proof. This is an analogue of Lemma 7.8 and can be proved by properly modifying the
arguments in Lemma 7.8. For clarity, we also provide a proof in the following.
By Lemma 7.10, there exists a constant C0(1) such that
u(t;x) U(t 2;x) 8x ct C0(1); t 2:
Observe that there are t > 2 and > 2 +l such that
u(t ;x) (1 )U(t ( l );x) 8x ct C0(1):
Thus,
u(t ;x) (1 )U(t +l ;x) 8x2RN:
By Lemma 7.5,
u(t;x) (1 e (t t ))U(t +l e (t t );x) 8t t ; x2RN:
Similarly, by Lemma 7.10, there exists a constant C0(1) such that
u(t;x) U(t+ 2;x) 8x ct C0(1); t 2:
Observe that there are t+ > 0 and + > 2 +l such that
u(t+;x) (1 + )U(t+ + + l ;x) 8x ct0 C0(1)
118
By Lemma 7.5 again,
u(t;x) (1 + e (t t+))U(t+ + + l e (t t+);x) 8t t+; x2RN:
The lemma then follows.
Lemma 7.12. Let > 0, t1 > 0; and M 2R be given. Suppose that w ( ;x;t1) are the
solution of (1.2) for t 0 with the initial conditions
w (0;x;t1) = U(t1 ;x)&(x ct1 M) +U(t1 2 ;x)(1 &(x ct1 M)) 8x2RN;
where &(s) = 0 for s 0 and &(s) = 1 for s> 0. Then
w+(1;x;t1) (1 + )U(t1 + 1 + 2 3l )
w (1;x;t1) (1 )U(t1 + 1 2 + 3l );
for all x ct1 M +c and 0 < 1.
Proof. This is an analogue of Lemma 7.9 and can be proved by properly modifying the
arguments in Lemma 7.9. Again, for clarity, we provide a proof in the following.
First we consider w . Note that
w (0;x;t1) = U(t1 2 ;x) 8x ct1 M;
and
w (0;x;t1) = U(t1 ;x) >U(t1 2 ;x) 8x ct1 >M:
By Proposition 3.1,
w (1;x;t1) >U(t1 + 1 2 ;x) 8x2RN:
By Lemma 7.6, for 0 < 1 < 0,
U(t1 + 1 2 ;x) (1 )U(t1 + 1 2 + 3l ;x) 8x2RN; x c(t1 + 1) M1( 1):
119
By arguments similar to those in Lemma 7.9, we can prove that
w (1;x;t1) > (1 )U(t1 + 1 2 + 3l ;x) 8x2RN; x c(t1 + 1)2[ M1( 1);M]
provided that 0 < 1. We then have
w (1;x;t1) = w(1;x) +U(1 +t1 2 ;x)
U(1 +t1 2 + 3l ;x)
(1 )U(1 +t1 2 + 3l ;x)
for x2RN with x c(1 +t1) M provided that 0 < 1. The statement for w then
follows.
Similarly, we can prove the case of w+. Hence, the lemma follows.
Proof of Theorem J. First of all, let
A+0 :=f 0jlim sup
t!1
sup
x2RN
u(t;x)
U(t+ 2 ;x) 1g
and
A 0 :=f 0jlim inft!1 inf
x2RN
u(t;x)
U(t 2 ;x) 1g:
De ne
+0 := inff j 2A+0g; 0 := inff j 2A 0g:
By Lemma 7.11, A 0 6=;. Hence 0 are well de ned.
By the similar arguments as in the proof of 2 A in Theorem I, we have that
0 2A 0 .
It then su ces to prove that 0 = 0. This can be proved again by the similar arguments
as in the proof of = 0 in Theorem I. For clarity, we provide a proof for the case of +0 .
We prove +0 = 0 by contradiction. Suppose for the contrary that +0 > 0. Then by the
de nition of +0 , for any given ^ > 0, there exists t0 > 0 such that
120
u(t0;x) U(t0 + 2 +0 ;x) + ^ ; 8x2RN:
Let w+(t;x;t0) be the solution of (1.2) for t 0 with initial condition given by
w+(0;x;t0) = U(t0 + +0 ;x)&(x ct0 M) + (1 &(x ct0 M))U(t0 + 2 +0 ;x);
where M = C0( +02 ) +c +0 .
Then, w+(0;x;t0) = U(t0 + 2 +0 ;x) for x ct0 M, which implies
u(t0;x) w+(0;x;t0) + ^ ; 8x ct0 M:
On the other hand, by Lemma 7.10,
u(t0;x) U(t0 + +0 ;x) 8x c(t0 + +0 ) C0(
+
0
2 ):
Hence
u(t0;x) U(t0 + +0 ;x); 8x ct0 M:
Therefore,
u(t0;x) w+(0;x;t0) + ^ ; 8x2RN:
Note that there is K > 0 such that w+(t;x;t0) + ^ eKt is a super-solution of (1.2) for
t2[0;1] provided that 0 < ^ 1. By Proposition 3.1,
u(t0 + 1;x) w+(1;x;t0) + ^ eK; 8x2RN:
Thus, by Lemma 7.12,
u(t0 + 1;x) (1 + )U(t0 + 1 + 2 +0 3l ;x) + ^ eK; 8x ct0 M +c
provided that 0 < ^ 1 and 0 < 1. Choose ^ to be su ciently small, we have
u(t0 + 1;x) (1 + 2 )U(t0 + 1 + 2 +0 3l ;x); 8x ct0 M +c:
121
On the other hand, by Lemma 7.10 again,
u(t0 + 1;x) U(t0 + 1 + +0 ;x) 8x c(t0 + 1 + +0 ) C0(
+
0
2 ):
This implies that for 0 < 1,
u(t0 + 1;x) < (1 + 2 )U(t0 + 1 + 2 +0 3l ;x) 8x ct0 c+c +0 +C0(
+
0
2 ) = c+M:
Hence, for 0 < 1,
u(t0 + 1;x) (1 + 2 )U(t0 + 1 + 2 +0 3l ;x); 8x2RN:
By Proposition 3.1 and Lemma 7.5,
u(t+t0 + 1;x) (1 + 2 e t)U(t+t0 + 1 + 2 +0 2l l e t); 8t> 0;x2RN:
Letting t!1, we have +0 l 2A+ for 0 < 1, which contradicts the de nition of +0 .
Hence we must have +0 = 0.
122
Chapter 8
Concluding Remarks, Open Problems, and Future Plan
In this dissertation, we studied the spatial spread and front propagation dynamics of
monostable equations with nonlocal dispersal in spatially periodic habitats. We rst estab-
lished a general principal eigenvalue theory for spatially periodic nonlocal dispersal operators.
More precisely, we investigated the following eigenproblem,
Z
RN
k(y x)v(y)dy v(x) +a(x)v(x) = v; v2Xp;
where a(x)2Xp, and provided some su cient conditions for the existence of principal eigen-
value and its associated positive eigenvector. The principal eigenvalue theory established in
this dissertation provides an important tool for the study of nonlinear evolution equations
with nonlocal dispersal and is also of great interest in its own.
Applying the principal eigenvalue theory for nonlocal dispersal operators and compari-
son principle for sub- and super-solutions, we obtained the existence, uniqueness, and global
stability of spatially periodic positive stationary solutions to a general spatially periodic non-
local monostable equation. It should be pointed out that in [13], the authors also provided
some su cient conditions for the existence of a principal eigenfunction of some nonlocal
operators on some bounded or unbounded domain. Similar statements to Theorem C(1) are
proved in [13] for time independent nonlocal KPP equations. We learned the work [13] while
the paper [57] was almost nished. The proof of Theorem C(1) in this dissertation or [57] is
di erent from that in [13].
Applying the principal eigenvalue theory for nonlocal dispersal operators and compar-
ison principle for sub- and super-solutions, we proved the existence of a spatial spreading
123
speed of a general spatially periodic nonlocal equation in any given direction, which char-
acterizes the speed at which a species invades into the region where there is no population
initially in the given direction. Moreover, it is shown that spatial variation of the habitat
speeds up the spatial spread of the population.
We remark that though we used the principal eigenvalue theory for nonlocal disper-
sal operators in their proofs, the existence, uniqueness, and stability of spatially periodic
positive stationary solutions and the existence of spreading speeds are generic features for
nonlocal monostable equations in the sense that they are independent of the existence of the
principal eigenvalue of the linearized nonlocal dispersal operator at the trivial solution of the
monostable equation, which is of great biological importance.
Assuming the existence of the principal eigenvalues of certain nonlocal dispersal op-
erators related to the linearized nonlocal dispersal operator at the trivial solution of the
monostable equation, we showed that a spatially periodic nonlocal monostable equation has
in any given direction a unique stable spatially periodic traveling wave solution connecting
its unique positive stationary solution and the trivial solution with all propagating speeds
greater than the spreading speed in that direction. It should be pointed out that in [17], J.
Coville, J. D avila and S. Mart nez proved the the existence of the traveling wave solutions
for nonlocal dispersal with KPP nonlinearity for speed c c ( ). But they did not inves-
tigate the uniqueness and stability of the traveling wave solutions. We learned the work
[17], while I completed almost all the work of this dissertation with my adviser Dr W. Shen
and submitted the joint work [58]. We did not include the case with the speed c = c ( ).
But in our work, we further investigated the uniqueness and stability of the traveling wave
solutions. Since we did independently, the methods in [17] and [58]are also di erent. We
remark that in [17], the kernel is symmetric with bounded support and in [58], the kernel
is also supported on a bounded ball. In this dissertation, we extended the kernel to a more
general case.
124
Along the line of my dissertation, there are several important open problems. We discuss
the following three problems.
Open problem 1. In [17], assuming the existence of the principal eigenvalues of certain
nonlocal dispersal operators related to the linearized nonlocal dispersal operator at the trivial
solution of (1.2), the authors proved that (1.2) has a traveling wave solution in the given
direction of 2SN 1 with speed c = c ( ). It is an open question whether the traveling wave
solution in a given direction of 2SN 1 with speed c = c ( ) is unique and stable.
Among the main techniques in proving the existence of traveling wave solutions are
comparison principle and sub- and super-solutions. Recall that on the construction of sub-
or super- solutions, the positive principal eigenfunctions play important roles. We proved
the "monstable" feature of our equation and the existence of spreading speed no matter (H4)
is satis ed or not. However, we only proved the case under the assumption (H4) for traveling
wave solutions and in [17], the authors also assumed (H4). Then the following is an open
question.
Open problem 2. It also remains open whether a general spatially periodic monostable
equation with nonlocal dispersal in RN with N 3 has traveling wave solutions connecting
the spatially periodic positive stationary solution u+ and 0 and propagating with constant
speeds.
If we add the temporal variable t to the growth rate function f, the following problem
is of great biological interest and is very challenging mathematically.
Open problem 3. How about the spatial spread and front propagation dynamics of the
nonlocal monostable equations involving both space and temporal variations, which is modeled
by the following equation,
@u
@t =
Z
RN
k(y x)u(t;y)dy u(t;x) +u(t;x)f(t;x;u(t;x)); x2 ? (8.1)
125
As for my future research plan, here are some of the problems I attempt to study in the
near future.
I would like to continue my study on spatial spread and front propagation dynamics of
monostable stable equations with nonlocal dispersal, in particular, I plan to investigate the
open problems mentioned above.
I would like to investigate the front propagation dynamics of other types of evolution
equations with nonlocal dispersal arising in applied problems, including nonlocal evolution
equations with combustion type and bistable type nonlinearities.
I also would like to extend my study of evolution equations with deterministic inhomo-
geneity to equations with random inhomogeneity.
126
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