Principal Eigenvalue Theroy for Time Periodic Nonlocal Dispersal Operators
and Applications
by
Nar Singh Rawal
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
August 02, 2014
Keywords: Nonlocal dispersal, random dispersal, principal eigenvalue, principal spectrum
point, vanishing condition, lower bound, monostable equation, spatial spreading speed,
traveling wave solution
Copyright 2014 by Nar Singh Rawal
Approved by
Wenxian Shen, Chair, Professor,Department of Mathematics and Statistics
Yanzhao Cao, Professor of Mathematics
Narendra K Govil,Associate Chair and Alumni Professor of Mathematics
Georg Hetzer, Professor of Mathematics
Amnon J Meir, Professor of Mathematics
Abstract
The dissertation is concerned with the spectral theory, in particular, the principal eigen-
value theory for nonlocal dispersal operators with time periodic dependence, and its applica-
tions. Nonlocal and random dispersal operators are widely used to model di usion systems
in applied sciences and share many properties. There are also some essential di erences
between nonlocal and random dispersal operators, for example, a smooth random dispersal
operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may
not have a principal eigenvalue.
In this dissertation, we rst establish criteria for the existence of principal eigenvalues
of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic
type boundary conditions. Among others, it is shown that a time periodic nonlocal disper-
sal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance
is su ciently small, or the time average of the underlying media satis es some vanishing
condition with respect to the space variable at a maximum point or is nearly globally ho-
mogeneous with respect to the space variable. We also obtain lower bounds of the principal
spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding
time averaged problems.
Next, we discuss the applications of the established principal eigenvalue theory to the
existence, uniqueness, and stability of time periodic positive solutions to Fisher or KPP type
equations with nonlocal dispersal in periodic media. We prove that such equations are of
monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique
time periodic positive solution u+(t;x) which is globally asymptotically stable.
Finally, we discuss the application of the established principal eigenvalue theory to the
spatial spreading and front propagation dynamics of KPP equations with nonlocal dispersal
ii
in periodic media. We show that such an equation has a spatial spreading speed c ( ) in
the direction of any given unit vector . A variational characterization of c ( ) is given.
Under the assumption that the nonlocal dispersal operator associated to the linearization of
the monostable equation at the trivial solution 0 has a principal eigenvalue, we also show
that the monostable equation has a periodic traveling wave solution connecting u+( ; ) and
0 propagating in any given direction of with speed c>c ( ).
Key words. Nonlocal dispersal, random dispersal, principal eigenvalue, principal spectrum
point, vanishing condition, lower bound, monostable equation, spatial spreading speed, trav-
eling wave solution.
Mathematics subject classi cation. 35K55, 35K57, 45C05, 45M15, 45M20, 47G10,
92D25.
iii
Acknowledgments
I feel pleasure by having the opportunity to discover the covered feeling with few words
for the people, department and university. Wenxian Shen, my advisor and Professor has
been the most in uenced person in my academic life. She taught me in every possible way
how to grow academically. Besides being the PhD advisor, she has been the instructor of
several advanced mathematical courses . Finding the most suitable word which correspond
the meaning of covered feeling may not be possible for me. All I can say is, the author is
greatly indebted to her and is very thankful to her for all kind of help, support, immotionally
and spritually. " You cannot convince others unless you are convinced and convincing idea
cannot be acheived without intensively pertinent thinking on the subject" is the great lesson
I have learned from her. Thank you so much for not only assigning me well posed problem
having outcome but also helping me to get the solution.
Among the other people having big academic shares in my academic life are Dr. Cao,
Dr. Govil, Dr. Hetzer and Dr. Meir . I had the opportunity to take classes under Dr.
Govil, Dr. Hetzer and Dr. Meir. All of them are great professors with kind hearted and
eleemosynary and benevolent nature having unique style of teaching. I feel myself lucky that
I have them all in my advisory committee.
Other people deserving to be on the acknowledgement are all the people from the o ce of
Mathematics and Statistics Department. In the period of my graduate study, the Department
of Mathematics and Statistics has seen two chairs. Dr. Smith and Dr. Tam. Both of them
are great academicians as well as great administrators. In addition, the Department has
diligent and helpful secretaries, Gwen, Carolyn, Lori. You need not to wait any longer if
you need something from the department, for examples, letters, stationary, books, etc. All
of them have been doing their job sincerely and diligently.
iv
Dr. Sushil Adhikari also deserved to be thanked from the author of this dissertation.
Despite being so hectic with academic duties and responsibilities , he agreed to be outside
reader for my thesis. I cannot forget such help rendered by Dr. Adhikari.
My wife Gomati and two sons, Sudarshan and Hemant have been my great source of
inspiration. I would like to mention names of my parents, Dan Singh Rawal and Belu Devi
Rawal. Although both of them cannot see their son?s graduation , I beleive that there must
be somethig by which they get the message about their son. I would also like to mention
one more name from my extended family, Tika Singh Rawal, my elder brother who helped
me immensely during my college education.
In addition to the people mentioned above, I would also like to thank the NSF for the
nancial support (NSF-DMS-0907752).
Last but the most importantly, my sincere thanks go to my creator GOD(JAGATNNATH
BABA) for sending me to one of the World?s most beautiful place Auburn for my graduate
study. This small university town with huge diversity has been able to draw my special at-
tention. I would like to express my gratefulness from the bottom of my heart to the almighty
that I was provided this golden opportunity to stay in great place (the loveliest village on
the plains) with great people during my graduate study.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Notations, De nitions, and Main Results . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Principal eigenvalues and principal spectrum points . . . . . . . . . . . . . . 11
2.2 Time periodic positive solutions of nonlocal KPP equations . . . . . . . . . . 13
2.3 Spatial spreading speeds of time and space periodic KPP equations . . . . . 15
2.4 Traveling wave solutions of time and space periodic KPP equations . . . . . 18
3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Basic properties for solutions of nonlocal evolution equations . . . . . . . . . 21
3.2 Basic properties of principal eigenvalues and principal spectrum points of
nonlocal dispersal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Principal Eigenvalue and Principal Spectrum Point Theory . . . . . . . . . . . . 40
4.1 Proofs of Theorems A-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Other important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Time Periodic Positive Solutions of Nonlocal KPP Equations in Periodic Media 57
6 Spatial Spreading Speed of Nonlocal KPP Equations in Periodic Media . . . . . 64
7 Traveling Wave Solutions of Nonlocal KPP Equations in Periodic Media . . . . 76
7.1 Sub- and super-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.2 Traveling wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vi
Chapter 1
Introduction
Both random dispersal evolution equations and nonlocal dispersal evolution equations
are widely used to model di usive systems in applied sciences. Classically, one assumes
that the internal interaction of organisms in a di usive system is in nitesimal or the internal
dispersal is random, which leads to a di usion operator, e.g., u as dispersal operator. Many
di usive systems in real world exhibit long range internal interaction or dispersal, which can
be modeled by nonlocal dispersal operators such asRRN (y x)u(t;y)dy u(t;x), here ( ) is
a convolution kernel supported on the ball centered at the origin with radius r, the interaction
range. As a basic technical tool for the study of nonlinear evolution equations with random
and nonlocal dispersals, it is of great importance to investigate aspects of spectral theory for
random and nonlocal dispersal operators.
This dissertation is devoted to the study of principal eigenvalues of the following three
eigenvalue problems associated to nonlocal dispersal operators with time periodic depen-
dence, (
ut + 1[RD (y x)u(t;y)dy u(t;x)] +a1(t;x)u = u; x2 D
u(t+T;x) = u(t;x)
(1.1)
where D RN is a smooth bounded domain and a1(t;x) is a continuous function with
a1(t+T;x) = a1(t;x),
( u
t + 2[
R
D (y x)(u(t;y) u(t;x))dy] +a2(t;x)u = u; x2 D
u(t+T;x) = u(t;x)
(1.2)
1
where D RN is as in (1.1) and a2(t;x) is a continuous function with a2(t+T;x) = a2(t;x),
and (
ut + 3[RRN (y x)u(t;y)dy u(t;x)] +a3(t;x)u = u; x2RN
u(t+T;x) = u(t;x+pjej) = u(t;x); x2RN
(1.3)
where pj > 0, ej = ( j1; j2; ; jN) with jk = 1 if j = k and jk = 0 if j6= k, and a3(t;x)
is a continuous function with a3(t+T;x) = a3(t;x+pjej) = a3(t;x), j = 1;2; ;N. ( ) in
(1.1)-(1.3) is a nonnegative C1 with compact support, (0) > 0, and RRN (z)dz = 1.
This dissertation is also devoted to the applications of the principal eigenvalue theory
for (1.1)-(1.3) to be developed.
The eigenvalue problems (1.1), (1.2), and (1.3) can be viewed as the nonlocal dispersal
counterparts of the following eigenvalue problems associated to random dispersal operators,
8
>><
>>:
ut + 1 u+a1(t;x)u = u; x2D
u(t+T;x) = u(t;x); x2D
u = 0; x2@D;
(1.4)
8>
><
>>:
ut + 2 u+a2(t;x)u = u; x2D
u(t+T;x) = u(t;x); x2D
@u
@n = 0; x2@D;
(1.5)
and (
ut + 3 u+a3(t;x)u = u; x2RN
u(t+T;x) = u(t;x+pjej) = u(t;x); x2RN;
(1.6)
respectively. It is in fact proved in [53] that the principal eigenvalues of (1.4), (1.5), and (1.6)
can be approximated by the principal spectrum points of (1.1), (1.2), and (1.3) with properly
rescaled kernels, respectively (see De nition 2.1 for the de nition of principal spectrum
points of (1.1), (1.2), and (1.3)). We may then say that (1.1), (1.2), and (1.3) are of the
Dirichlet type boundary condition, Neumann type bounday condition, and periodic boundary
condition, respectively. The reader is referred to [8], [9], and [53] about the approximations
of the initial value problems of the random dispersal operators associated to (1.4), (1.5), and
2
(1.6) by the initial value problems of the nonlocal dispersal operators with properly rescaled
kernels associated to (1.1), (1.2), and (1.3), respectively.
The eigenvalue problems (1.4), (1.5), and (1.6), in particular, their associated principal
eigenvalue problems, are well understood. For example, it is known that there is R;1 2R
such that R;1 is an isolated algebraic simple eigenvalue of (1.4) with a positive eigenfunction,
and for any other eigenvalues of (1.4), Re R;1 ( R;1 is called the principal eigenvalue
of (1.4)) (see [22]).
The principal eigenvalue problem for time independent nonlocal dispersal operators
with Dirichlet type, or Neumann type, or periodic boundary condition has been recently
studied by many people (see [11], [18], [23], [30], [49], [52], and references therein) and is
quite well understood now. For example, the following criteria for the existence of principal
eigenvalues for nonlocal dispersal operators are established in [49] and [52] ([49] is on the
periodic boundary condition case and [52] is on Dirichlet type and Neumann type boundary
conditions) (see De nition 2.1 for the de nition of principal eigenvalues of nonlocal dispersal
operators),
(i) If a1(t;x) a1(x) (resp. a2(t;x) a2(x), a3(t;x) a3(x)) and (z) = 1 N ~ (z ) for some
> 0 and ~ ( ) with ~ (z) 0, supp(~k) = B(0;1) :=fz2RNjkzk< 1g, and RRN ~ (z)dz = 1,
then (1.1) (resp. (1.2), (1.3)) admits a principal eigenvalue provided that is su ciently
small.
(ii) If a1(t;x) a1(x) (resp. a2(t;x) a2(x), a3(t;x) a3(x)) is CN and there is some
x0 2 Int(D) (resp. x0 2 Int(D), x0 2 RN) satisfying that a1(x0) = maxx2 Da1(x) (resp.
2RD (y x0)dy+a2(x0) = maxx2 D( 2RD (y x)dy+a2(x)), a3(x0) = maxx2RN a3(x))
and the partial derivatives of a1(x) (resp. 2RD (y x)dy+a2(x), a3(x)) up to order N 1
at x0 are zero, then (1.1) (resp. (1.2), (1.3)) admits a principal eigenvalue.
(iii) If a1(t;x) a1(x) (resp. a2(t;x) a2(x), a3(t;x) a3(x)) and maxx2 Da1(x)
minx2 Da1(x) < 1 infx2 DRD (y x)dy (resp. maxx2 Da2(x) minx2 Da2(x) < 2 infx2 DRD (y
3
x)dy, maxx2RN a3(x) minx2RN a3(x) < 3), then (1.1) (resp. (1.2), (1.3)) admits a princi-
pal eigenvalue.
It should be pointed out that [30] contains some results similar to (i) in the Dirichlet
type boundary condition case and [11] contains some results similar to (ii). It should also
be pointed out that a nonlocal dispersal operator may not have a principal eigenvalue (see
[49] for an example), which reveals some essential di erence between nonlocal and random
dispersal operators. Methologically, due to the lack of regularity and compactness of the
solutions of nonlocal evolution equations, some di culties, which do not arise in the study
of spectral theory of random dispersal operators, arise in the study of spectral theory of
nonlocal dispersal operators.
Regarding nonlocal dispersal operators with time periodic dependence, in [28], the au-
thors studied the existence of principal eigenvalues of (1.1) in the case that N = 1. In [28]
and [48], the in uence of temporal variation on the principal eigenvalue of (1.1) (if exists) is
investigated. In general, the understanding to the principal eigenvalue problems associated
to (1.1), (1.2), and (1.3) is very little.
The rst objective of this dissertation is to develop criteria for the existence of principal
eigenvalues of (1.1), (1.2), and (1.3) and to explore fundamental properties of principal
eigenvalues of (1.1), (1.2), and (1.3). Many existing results on principal eigenvalues of
time independent and some special time periodic nonlocal dispersal operators are extended
to general time periodic nonlocal dispersal operators. For example, the following result is
established in this dissertation, which extends (ii) in the above for time independent nonlocal
dispersal operators to time periodic ones,
If a1(t;x) (resp. a2(t;x), a3(t;x)) is in CN in x and there is some x0 2Int(D) (resp. x0 2
Int(D), x0 2RN) such that that ^a1(x0) = maxx2 D ^a1(x) (resp. RD (y x0)dy + ^a2(x0) =
maxx2 D ( RD (y x)dy + ^a2(x), ^a3(x0) = maxx2RN ^a3(x)) and the partial derivatives of
^a1(x) (resp. RD (y x)dy + ^a2(x), ^a3(x)) up to order N 1 at x0 are zero, then (1.1)
4
(resp. (1.2), (1.3)) admits a principal eigenvalue, where ^ai(x) is the time average of ai(t;x)
(i = 1;2;3) (see (2.1) for the de nition of ^ai( ) for i = 1;2;3).
The reader is referred to Theorems A-C in section 2 for the principal eigenvalue theories
established in this dissertation for general time periodic nonlocal dispersal operators.
The second objective of this dissertation is to consider applications of the established
principal theories to the following time periodic KPP type or Fisher type equations with
nonlocal dispersal,
ut = 1[
Z
D
(y x)u(t;y)dy u(t;x)] +uf1(t;x;u); x2 D; (1.7)
ut = 2[
Z
D
(y x)(u(t;y) u(t;x))dy] +uf2(t;x;u); x2 D; (1.8)
and (
ut = 3[RRN (y x)u(t;y)dy u(t;x)] +uf3(t;x;u); x2RN
u(t;x+pjej) = u(t;x); x2RN;
(1.9)
where fi(t;x) (i = 1;2;3) are C1 functions, fi(t + T;x;u) = fi(t;x;u) (i = 1;2;3), f3(t;x +
pjej;u) = f3(t;x;u) (j = 1;2; ;N), and fi(t;x;u) < 0 for u 1 and @ufi(t;x;u) < 0 for
u 0 (i = 1;2;3).
Equations (1.7), (1.8), and (1.9) are the nonlocal counterparts of the following reaction
di usion equations, (
ut = 1 u+uf1(t;x;u); x2D
u(t;x) = 0; x2@D;
(1.10)
(u
t = 2 u+uf2(t;x;u); x2D
@u
@n = 0; x2@D;
(1.11)
and (
ut = 3 u+uf3(t;x;u); x2RN
u(t;x+pjej) = u(t;x); x2RN;
(1.12)
respectively (see [53] for the approximations of the solutions of (1.7), (1.8), and (1.9) to
(1.10), (1.11), and (1.12), respectively).
5
Equations (1.7)-(1.9) and (1.10)-(1.12) are widely used to model population dynamics
of species exhibiting nonlocal internal interactions and random internal interactions, respec-
tively. Thanks to the pioneering works of Fisher ([17]) and Kolmogorov, Petrowsky, Piscunov
([31]) on the following special case of (1.12),
ut = uxx +u(1 u); x2R; (1.13)
(1.7)-(1.9) and (1.10)-(1.12) are referred to as Fisher type or KPP type equations.
One of the central problems for (1.7)-(1.9) and (1.10)-(1.12) is about the existence,
uniqueness, and stability of positive time periodic solutions. This problem has been exten-
sively studied and is well understood for (1.10)-(1.12). For example, it is known that (1.10)
exhibits the following monostable feature: if the trivial solution u 0 is a linearly unstable
solution of (1.10), then (1.10) has a unique stable time periodic positive solution. Again,
some di culties, which do not arise in the study of (1.10)-(1.12), aries in the study of (1.7)-
(1.9) due to the lack of compactness and regularities of the solutions of nonlocal dispersal
evolution equations. In [51], the authors proved that time independent KPP equations with
nonlocal dispersal also exhibit monostable feature (see also [2], [11] for the study of positive
stationary solutions of time independent KPP equations with nonlocal dispersal). But it is
hardly studied whether a general time periodic KPP equation with nonlocal dispersal is of
the monostable feature. In this dissertation, by applying the principal eigenvalue theories
for time periodic nonlocal dispersal operators to be established, we prove
A time periodic KPP equations with nonlocal dispersal is of the monostable feature, that
is, if u 0 is a linearly unstable solution of a time periodic KPP equation with nonlocal
dispersal, then the equation has a unique stable time periodic positive solution u+( ; ) (see
Theorem E in Section 2).
6
Consider (1.9) without the periodic condition u(t;x+piei) = u(t;x), that is,
ut = 3[
Z
RN
(y x)u(t;y)dy u(t;x)] +uf3(t;x;u); x2RN (1.14)
where f3(t;x;u) < 0 for u 1, @uf3(t;x;u) < 0 for u 0, and f3(t;x;u) is of certain
recurrent property in t and x. The spatial spreading and front propagation dynamics is also
among central problems. This problem has been studied by many people for the random
dispersal counterpart of (1.14) since the pioneering work by Fisher ([17]) and Kolmogorov,
Petrowsky, and Piscunov ([31]). Fisher in [17] 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 [31] by Kolmogorov, Petrowsky, and
Piscunov, that is, they proved that for any nonnegative solution u(t;x) of (1.13), if at time
t = 0, u is 1 near 1and 0 near1, 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). The number 2
is called the spatial spreading speed of (1.13) in literature.
The results of Fisher and Kolmogorov, Petrowsky, Piscunov [31] for (1.13) have been
extended by many people to quite general reaction di usion equations of the form,
ut = u+uf3(t;x;u); x2RN; (1.15)
where f3(t;x;u) < 0 for u 1, @uf3(t;x;u) < 0 for u 0, and f3(t;x;u) is of certain
recurrent property in t and x. For example, assume that f3(t;x;u) is periodic in t with
period T and periodic in xi with period pi (pi > 0, i = 1;2; ;N) (i.e. f3( + T; ; ) =
f3( ; + piei; ) = f3( ; ; ), ei = ( i1; i2; ; iN), ij = 1 if i = j and 0 if i 6= j, i;j =
1;2; ;N), and that u 0 is a linearly unstable solution of (1.15) with respect to periodic
perturbations. Then it is known that (1.15) has a unique positive periodic solution u+(t;x)
(u+(t + T;x) = u+(t;x + piei) = u+(t;x)) which is asymptotically stable with respect to
7
periodic perturbations and it has been proved that for every 2SN 1 :=fx2RNjkxk= 1g,
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 . Moreover, the minimal wave
speed c ( ) is of some important spreading properties. The reader is referred to [3], [4], [5],
[32], [33], [38], [39], [55], [56] and references therein for the above mentioned properties and
to [24], [37], [46], [47] for the extensions of the above results to the cases that f3(t;x;u) is
almost periodic in t and periodic in x and that f3(t;x;u) f3(t;u) is recurrent in t.
Recently, the spatial spreading and front propagation dynamics for (1.14) withf3(t;x;u) =
f3(x;u) has been studied by many authors. See, for example, [10], [12], [13], [14], [23], [34],
[36], [40], [49], [50], [51] for the study of the existence of spreading speeds and traveling
wave solutions of (1.14) connecting the trivial solution u = 0 and a nontrivial positive sta-
tionary solution in the case that f3(t;x;u) f3(x;u). However, in contrast to (1.15), the
spatial spreading and front propagation dynamics of (1.14) with both time and space pe-
riodic dependence or with general time and/or space dependence is much less understood.
The results on spatial spreading speeds and traveling wave solutions established in [33] and
[56] for quite general periodic monostable evolution equations cannot be applied to time and
space periodic nonlocal monostable equations because of the lack of certain compactness
of the solution operators for such equations. In this dissertation, by applying the principal
eigenvalue theories for time periodic nonlocal dispersal operators to be established, we obtain
For any given unit vector 2RN, (1.14) has a spatial spreading speed c ( ) in the direction
of . Moreover, some variational characterization for c ( ) is given and the spreading speed
c ( ) is of some important spreading features (see Theorems G and H for detail).
If for given 2RN with k k= 1, the following eigenvalue problem
( u
t +
R
RN e
(y x) (y x)u(t;y)dy u(t;x) +a0(t;x)u(t;x) = u(t;x)
u(t+T;x) = u(t;x+piei) = u(t;x)
(1.16)
8
has a principal eigenvalue 0( ; ;a0) for each > 0, where a0(t;x) = f3(t;x;0), then for any
c>c ( ), (1.14) has a (periodic) traveling wave solution u(t;x) = (x ct;t;ct) connecting
u+ and 0 (see Theorem I for detail).
The results stated above cover most of the results in literature whenf3(t;x;u) f3(x;u).
It should be pointed out that if (1.16) has no principal eigenvalue for some > 0, it remains
open whether (1.14) has a traveling wave solution connecting u+( ; ) and 0 in the direction of
with speed c>c ( ) (this remains open even when f3(t;x;u) f3(x;u) is time independent
but space periodic).
Nonlocal evolution equations have been attracting more and more attention due to the
presence of nonlocal interaction in many di usive systems in applied sciences. The reader is
referred to [7], [10], [12], [14], [16], [18], [19], [30], [34], [36], [40], [48], [50], etc., for the study
of various aspects of nonlocal dispersal evolution equations.
The rest of the dissertation is organized as follows. In Chapter 2, we introduce standing
notations and de nitions and state the main results of the dissertation. We present basic
properties needed in the proofs of the main results in Chapter 3. The principal eigenvalue
theory is developed in Chapter 4. The chapter 5 is about time periodic positive solutions of
nonlocal KPP equations in periodic media. In Chapters 6 and 7, the spatial spreading speeds
and traveling wave solutions of nonlocal KPP equations in periodic media are presented,
respectively.
9
Chapter 2
Notations, De nitions, and Main Results
This chapter begins with standing notations that are used in this chapter and beyond.
Following the notations, the de nitions of principal spectrum points and principal eigen-
values of (1.1), (1.2), and (1.3) are given. We then state the main results concerning the
existence of principal eigenvalues, its applications to time periodic KPP equations with non-
local dispersal.
Let
X1 =X2 =fu2C(R D;R)ju(t+T;x) = u(t;x)g
with norm kukXi = supt2R;x2 Dju(t;x)j (i = 1;2),
X3 =fu2C(R RN;R)ju(t+T;x) = u(t;x+piei) = u(t;x)g
with norm kukX3 = supt2R;x2RNju(t;x)j, and
X+i =fu2Xiju 0g
(i = 1;2;3).
For given ai2Xi, let Li(ai) :D(Li(ai)) Xi!Xi be de ned as follows,
(L1(a1)u)(t;x) = ut(t;x) + 1[
Z
D
(y x)u(t;y)dy u(t;x)] +a1(t;x)u(t;x);
(L2(a2)u)(t;x) = ut(t;x) + 2
Z
D
(y x)(u(t;y) u(t;x))dy +a2(t;x)u(t;x);
10
and
(L3(a3)u)(t;x) = ut(t;x) + 3[
Z
RN
(y x)u(t;y)dy u(t;x)] +a3(t;x)u(t;x):
De nition 2.1. Let
s(Li;ai) = supfRe j 2 (Li(ai))g
for i = 1;2;3. Then, s(Li;ai) is called the principal spectrum point of L(ai) (i = 1;2;3).
If s(Li;ai) is an isolated eigenvalue of L(ai) with a positive eigenfunction (i.e. 2X+i ),
then s(Li;ai) is called the principal eigenvalue of Li(ai) or it is said that Li(ai) has a principal
eigenvalue (i = 1;2;3).
For given 1 i 3 and a2Xi, let
^ai(x) = 1T
Z T
0
ai(t;x)dt; (2.1)
bi(x) =
(
i for i = 1;3;
iRD (y x)dy for i = 2;
(2.2)
and
Di =
(D for i = 1;2;
[0;p1] [0;p2] [0;pN] for i = 3:
(2.3)
2.1 Principal eigenvalues and principal spectrum points
Our main results on the principal spectrum points and principal eigenvalues of nonlocal
dispersal operators can then be stated as follows.
Theorem A. (Necessary and su cient condition)
Let 1 i 3 be given. Then, s(Li;ai) is the principal eigenvalue of Li(ai) i s(Li;ai) >
maxx2 Di(bi(x) + ^ai(x)).
11
Theorem B. (Su cient conditions)
Let 1 i 3 be given.
(1) Suppose that (z) = 1 N ~ (z ) for some > 0 and ~ ( ) with ~ (z) 0, supp(~ ) =
B(0;1) := fz2RNjkzk< 1g, and RRN ~ (z)dz = 1. Then the principal eigenvalue of
Li(ai) exists for 0 < 1.
(2) The principal eigenvalue of Li(ai) exists if ai(t;x) is in CN in x, there is some x0 2
Int(Di) in the case i = 1;2 and x0 2D3 in the case i = 3 satisfying that bi(x0)+^ai(x0) =
maxx2 Di(bi(x) + ^ai(x)), and the partial derivatives of bi(x) + ^ai(x) up to order N 1
at x0 are zero.
(3) The principal eigenvalue of Li(ai) exists if
max
x2 Di
^ai(x) min
x2 Di
^ai(x) < i inf
x2 Di
Z
Di
(y x)dy
in the case i = 1;2 and
max
x2 Di
^ai(x) min
x2 Di
^ai(x) < i
in the case i = 3.
Theorem C. (In uence of temporal variation)
For given 1 i 3, s(Li;ai) s(Li;^ai).
Corollary D. If s(Li;^ai) is the principal eigenvalue of Li(^ai), then s(Li;ai) is the principal
eigenvalue of Li(ai).
Proof. Assume that s(Li;^ai) is the principal eigenvalue of Li(^ai). Then by Theorem A,
s(Li;^ai) > max
x2 Di
(bi(x) + ^ai(x)):
12
This together with Theorem C implies that
s(Li;ai) > max
x2 Di
(bi(x) + ^ai(x)):
Then by Theorem A again, s(Li;ai) is the principal eigenvalue of Li(ai).
Observe that when ai(t;x) ai(x) (i = 1;2;3), Theorems A and B recover the exist-
ing results for time independent nonlocal dispersal operators (see [49], [52], and references
therein). Theorem B (2) extends a result in [28] for the case i = 1 and N = 1 to time
periodic nonlocal dispersal operators in higher space dimension domains. In the case i = 1
and both s(L1;a1) and s(L1;^a1) are eigenvalues of L1(a1) and L1(^a1), it is shown in [28] that
s(L1;a1) s(L1;^a1). Theorem C extends this result to general time periodic nonlocal dis-
persal operators and shows that temporal variation does not reduce the principal spectrum
point of a general time periodic nonlocal dispersal operator.
Theorems A-C and Corollary D establish some fundamental principal eigenvalue theory
for general time periodic nonlocal dispersal operators and provide a basic tool for the study
of nonlinear evolution equations with nonlocal dispersal. In the following, we consider their
applications to the study of the asymptotic dynamics of (1.7)-(1.9).
2.2 Time periodic positive solutions of nonlocal KPP equations
Let
X1 = X2 =fu2C( D;R)g
with norm kukXi = supx2 Dju(x)j (i = 1;2),
X3 =fu2C(RN;R)ju(x+pjej) = u(x)g
13
with norm kukX3 = supx2RNju(x)j, and
X+i =fu2Xiju 0g;
(i = 1;2;3) and
X++i =
(u2X+
i ju(x) > 0 8x2 D; i = 1;2
u2X+i ju(x) > 0 8x2RN; i = 3:
.
By general semigroup theory, for any s2R and u0 2X1 (resp. u0 2X2, u0 2X3),
(1.7) (resp. (1.8), (1.9)) has a unique (local) solution u1(t;x;s;u0) (resp. u2(t;x;s;u0),
u3(t;x;s;u0)) withu1(s;x;s;u0) = u0(x) (resp. u2(s;x;s;u0) = u0(x), u3(s;x;s;u0) = u0(x))
(see Proposition 3.1). Moreover, if u0 2X+i , then ui(t;x;s;u0) exists and ui(t; ;s;u0)2X+i
for all t s (i = 1;2;3) (see Proposition 3.3).
Theorem E. (Existence, uniqueness, and stability of time periodic positive solutions)
Let ai(t;x) = fi(t;x;0) (i = 1;2;3). If s(L1;ai) > 0 (resp. s(L2;a2) > 0, s(L3;a3) > 0),
then (1.7) (resp. (1.8), (1.9)) has a unique time periodic solution solution u 1(t; ) 2X++1
(resp. u 2(t; )2X++2 , u 3(t; )2X++3 ). Moreover, for any u0 2X+i nf0g,
kui(t; ; 0;u0) u i(t; )kXi !0
as t!1 (i = 1;2;3).
Corollary F. Let ai(t;x) = fi(t;x;0) (i = 1;2;3). If s(L1;^a1) > 0 (resp. s(L2;^a2) > 0,
s(L3;^a3) > 0), then (1.7) (resp.( 1.8), (1.9)) has a unique time periodic solution solution
u 1(t; )2X++1 (resp. u 2(t; )2X++2 , u 3(t; )2X++3 ). Moreover, for any u0 2X+i nf0g,
kui(t; ; 0;u0) u i(t; )kXi !0
as t!1 (i = 1;2;3).
14
Proof. Assume s(L1;^a1) > 0 (resp. s(L2;^a2) > 0, s(L3;^a3) > 0). By Theorem C, s(L1;ai) >
0 (resp. s(L2;a2) > 0, s(L3;a3) > 0). The corollary then follows from Theorem E.
2.3 Spatial spreading speeds of time and space periodic KPP equations
For simplicity in notation, when considering the spatial spreading and front propagation
dynamics of (1.14), we drop the sub-index 3, that is, we write (1.14) as
@u
@t =
Z
RN
(y x)u(t;y)dy u(t;x) +u(t;x)f(t;x;u(t;x)); x2RN; (2.4)
where ( ) is as in (1.14) andf(t;x;u) is periodic intandxand satis es proper monostablility
assumptions. More precisely, let (H0) stands for the following assumption.
(H0) f(t;x;u) is C1 in (t;x;u)2R RN R, and f( +T; ; ) = f( ; +piei; ) = f( ; ; ),
ei = ( i1; i2; ; iN), ij = 1 if i = j and 0 if i6= j, i;j = 1;2; ;N.
Let
Xp =fu2C(R RN;R)ju( +T; ) = u( ; +piei) = u( ; ); i = 1; ;Ng (2.5)
with norm kukXp = sup(t;x)2R RNju(t;x)j (note that Xp =X3), and
X+p =fu2Xpju(t;x) 0 8(t;x)2R RNg: (2.6)
Let I be the identity map on Xp, and K, a0( ; )I :Xp!Xp be de ned by
(Ku)(t;x) =
Z
RN
(y x)u(t;y)dy; (2.7)
(a0( ; )Iu)(t;x) = a0(t;x)u(t;x); (2.8)
15
where a0(t;x) = f(t;x;0). Let ( @t +K I +a0( ; )I) be the spectrum of @t +K I +
a0( ; )I acting on Xp. The monostablility assumptions are then stated as follows:
(H1) @f(t;x;u)@u < 0 for t2R, x2RN and u2R and f(t;x;u) < 0 for t2R, x2RN and
u 1.
(H2) u 0 is linearly unstable in Xp, that is, 0(a0) > 0, where 0(a0) := supfRe j 2
( @t +K I +a0( ; )).
Let
X =fu2C(RN;R)ju is uniformly continuous and boundedg (2.9)
with supremum norm and
X+ =fu2Xju(x) 0 8x2RNg: (2.10)
By general semigroup theory, for any u0 2X, (2.4) has a unique solution u(t;x;u0) with
u(0;x;u0) = u0(x). By comparison principle, if u0 2X+, then u(t; ;u0) exists for all t 0
and u(t; ;u0)2X+ (see Proposition 3.3 for detail).
By Theorem E, (H1) and (H2) imply that (2.4) has exactly two time periodic solutions in
X+p , u = 0 and u = u+(t;x), and u = 0 is linearly unstable and u = u+(t;x) is asymptotically
stable with respect to positive perturbations in X+p , where
Xp =fu2C(RN;R)ju( +pei) = u( )g (2.11)
with maximum norm (note that Xp = X3) and
X+p =fu2Xpju(x) 0 8x2RNg: (2.12)
Hence (H1) and (H2) are called monostability assumptions.
16
For given 2SN 1 and 2R, let 0( ; ;a0) be the principal spectrum point of the
eigenvalue problem
( u
t +
R
RN e
(y x) (y x)u(t;y)dy u(t;x) +a0(t;x)u(t;x) = u(t;x)
u( ; )2Xp
(2.13)
(de ned as in De nition 2.1 with (y x) being replaced by e (y x) (y x)). Let X+( )
be de ned by
X+( ) =fu2X+j inf
x 1
u(x) > 0; sup
x 1
u(x) = 0g: (2.14)
De nition 2.2. For a given vector 2SN 1, let
C inf( ) =
n
cj8u0 2X+( ); lim sup
t!1
sup
x ct
ju(t;x;u0) u+(t;x)j= 0
o
and
C sup( ) =
n
cj8u0 2X+( ); lim sup
t!1
sup
x ct
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 (2.4) in the direction of . If c inf( ) =
c sup( ), we call c ( ) := cinf( ) the spreading speed of (2.4) in the direction of .
Theorem G. (Existence of spreading speeds) Assume (H1) and (H2). For any given 2
SN 1, c inf( ) = c sup( ) and hence the spreading speed c ( ) of (2.4) in the direction of
exists. Moreover,
c ( ) = inf >0 0( ; ;a0) ;
where a0(t;x) = f(t;x;0).
17
Theorem H. (Spreading features of spreading speeds) Assume (H1) and (H2).
(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) = 0:
(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) u+(t;x)) = 0
for every u0 2X+ satisfying u0(x) for all x2RN with jx j r.
(3) 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) = 0
for all c> sup 2SN 1 c ( ).
(4) Assume that 0 0 and r> 0,
lim inft!1 inf
kxk ct
(u(t;x;u0) u+(t;x)) = 0
for every u0 2X+ satisfying u0(x) for x2RN with kxk r.
2.4 Traveling wave solutions of time and space periodic KPP equations
De nition 2.3 (Traveling wave solution). (1) An entire solution u(t;x) of (2.4) is called
a traveling wave solution connecting u+( ; ) and 0 and propagating in the direction of
with speed c if there is a bounded function : RN R RN !R+ such that ( ; ; )
is Lebesgue measurable, u(t; ; ( ;0;z);z) exists for all t2R,
u(t;x) = u(t;x; ( ;0;0);0) = (x ct ;t;ct ) 8t2R; x2RN; (2.15)
18
u(t;x; ( ;0;z);z) = (x ct ;t;z +ct ) 8t2R; x;z2RN; (2.16)
lim
x ! 1
( (x;t;z) u+(t;x+z)) = 0; lim
x !1
(x;t;z) = 0 (2.17)
uniformly in (t;z)2R RN,
(x;t;z x) = (x0;t;z x0) 8x;x02RN with x = x0 ; (2.18)
and
(x;t+T;z) = (x;t;z +piei) = (x;t;z) 8x;z2RN: (2.19)
(2) A bounded function : RN R RN ! R+ is said to generate a traveling wave
solution of (2.4) in the direction of with speed c if it is Lebesgue measurable and
satis es (2.16) - (2.19).
Remark 2.4. Suppose that u(t;x) = (x ct ;t;ct ) is a traveling wave solution of (2.4)
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;t;x) (2.20)
for some : R R RN ! R satisfying ( ;t + T;z) = ( ;t;z + piei) = ( ;t;z),
lim ! 1 ( ;t;z) = u+(t;z), and lim !1 ( ;t;z) = 0 uniformly in (t;z) 2R RN. In
fact, let ( ;t;z) = (x;t;z x) for x2RN with x = . Observe that ( ;t;z) is well
de ned and has the above mentioned properties.
For convenience, we introduce the following assumption:
(H3) For every 2SN 1 and 0, 0( ; ;a0) is the principal eigenvalue of @t +K ;
I +a0( ; )I, where a0(t;x) = f(t;x;0).
We now state the main results of this section. For given 2SN 1 and c > c ( ), let
2(0; ( )) be such that
c = 0( ; ;a0) :
19
Let ( ; ; ) 2X+p be the positive principal eigenfunction of @t +K ; I + a0( )I with
k ( ; ; )kXp = 1.
Theorem I. (Existence of traveling wave solutions) Assume (H1)-(H3). For any 2SN 1
and c > c ( ), there is a bounded function : RN R RN ! R+ such that ( ; ; )
generates a traveling wave solution connecting u+( ; ) and 0 and propagating in the direction
of with speed c. Moreover, lim
x !1
(x;t;z)
e x ( ;t;x+z) = 1 uniformly in t2R and z2R
N.
20
Chapter 3
Basic Properties
In this chapter, we present basic properties to be used in the following chapters.
3.1 Basic properties for solutions of nonlocal evolution equations
In this section, we present some basic properties for solutions of (1.7)-(1.9) and linear
nonlocal evolution equations,
ut = 1[
Z
D
(y x)u(y)dy u(x)] +a1(t;x)u; x2 D; (3.1)
ut = 2[
Z
D
(y x)(u(y) u(x))dy] +a2(t;x)u; x2 D; (3.2)
and
ut = 3[
Z
RN
(y x)u(y)dy u(x)] +a3(t;x)u; x2RN; (3.3)
where ai2Xi (i = 1;2;3).
Throughout this chapter, i denotes any integer with 1 i 3, unless speci ed otherwise
and Xi, X+i , and Xi, X+i , X++i are as in section 2. Di is as in (2.3). For u1;u2 2Xi, we
de ne
u1 u2 (u1 u2) if u2 u1 2X+i (u1 u2 2X+i ):
For u1;u2 2Xi, we de ne
u1 u2 (u1 u2) if u2 u1 2X+i (u1 u2 2X+i );
and
u1 u2 (u1 u2) if u2 u1 2X++i (u1 u2 2X++i ):
21
Proposition 3.1. (1) For any u0 2X1 (resp. u0 2X2, u0 2X3) and s2R, (3.1) (resp.
(3.2), (3.3)) has a unique solution u(t; ;s;u0), denoted by 1(t;s)u0 (resp. 2(t;s)u0,
3(t;s)u0) with u(s;x;s;u0) = u0(x).
(2) For any u0 2X1 (resp. u0 2X2, u0 2X3) and s2R, (1.7) (resp. (1.8), (1.9)), has a
unique (local) solution u1(t; ;s;u0) (resp. u2(t; ;s;u0), u3(t; ;s;u0)) with u1(s;x;s;u0) =
u0(x) (resp. u2(s;x;s;u0) = u0(x), u3(s;x;s;u0) = u0(x)).
Proof. (1) We prove the existence of a unique solution of the initial value problem associated
to (3.1). The existence of unique solutions of the initial value problems associated to (3.2)
and (3.3) can be proved similarly.
Assume 0 s < t T. De ne K1 : X1 !X1 and A1(t) : X1 !X1 by (K1u)(x) =
1[RD (y x)u(y)dy u(x)] and (A1(t)u)(x) = a1(t;x)u(x). Then, K1 and for every t,
A1(t) are linear, bounded operators on X1. Assume A(t) := K1 + A1(t). Then, for every t,
0 t T, A(t) is a bounded linear operator on X1. The function t!A(t) is continuous in
the uniform operator topology. Then, by [[41], Chapter 5,Theorem 5.1], for every u0 2X1,
the initial value problem, du(t)dt = A(t)u(t); 0 s < t T with u(s) = u0 has a unique
classical solution u(t; ;s;u0):
(2) Write (1.7) as ut = A1u + g1(t;x;u) where A1u = 1RD (y x)u(y)dy u(x) and
g1(t;x;u) = u(x)f1(t;x;u): Then A1 is bounded linear operator and hence generates a C0
semigroup on X1 and g1 is continuous in t and Lipschitz continuos in u because of f1. By
[[41], Chapter 6, Theorem 1.4], for any u0 2X1, (1.7) has a unique local solution u1(t; ;s;u0)
with u1(s; ;s;u0) = u0( ).
The existence of unique solutions of the initial value problems associated to (1.7) and
(1.9) can be proved analogously.
De nition 3.2. A continuous function u(t;x) on [0; ) D is called a super-solution (or
sub-solution) of (1.7) if for any x2 D, u(t,x) is di erentiable on [0; ) and satis es that for
22
each x2 D,
@u
@t (or ) 1
hZ
D
(y x)u(t;y)dy u(t;x)
i
+u(t;x)f1(t;x;u)
for t2[0; ).
Super-solutions and sub-solutions of (1.8), (1.9), and (3.1)-(3.3) are de ned in an anal-
ogous way.
Proposition 3.3 (Comparison principle).
(1) If u1(t;x) and u2(t;x) are sub-solution and super-solution of (3.1) ( (resp. 3.2) ,
(3.3)) on [0;T), respectively, u1(0; ) u2(0; ), and u2(t;x) u1(t;x) 0 for
(t;x)2[0;T) Di 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 (1.7) (resp. (1.8), (1.9))
on [0;T), respectively, and u1(0; ) u2(0; ), then u1(t; ) u2(t; ) for t2[0;T).
(3) For every u0 2X+i , ui(t;x;s;u0) exists for all t s.
Proof. (1) We prove the case that u1(t;x) and u2(t;x) are sub-solution and super-solution
of (3.1). Other cases can be proved similarly.
Let u1(t;x) and u2(t;x) be sub-solution and super-solution of (3:1) respectively. De ne
v(t;x) = e t(u2(t;x) u1(t;x)) and p1 = 1 +a1(t;x). Then v satis es
@v
@t 1
Z
D
(y x)v(t;y)dy +p1(t;x)v(t;x); x2D:
Choose > 0 so large enough that p1(t;x) 0 for (t;x) 2 (0;T) D: We need to
prove v(t;:) 0 for t 2 (0;T): It su ces to prove v(t;:) 0 for t 2 (0;T0) where
T0 = minfT; 1k0+p0g, k0 = maxx2DRD (y x)dy and p0 = sup(t;x)2(0;T) Dp(t;x).
Suppose not. Then there exists (t0;x0) 2 (0;T0) D such that v(t0;x0) < 0: Let
vinf = inf(t;x)2(0;t0] Dv(t;x). Then vinf < 0: Choose the sequence (tn;xn)2(0;t0] D such
23
that v(tn;xn)!vinf as n!1: Then we have,
v(tn;xn) v(0;xn)
Z tn
0
[
Z
D
(y xn)v(t;y)dy +p1(t;xn)v(t;xn)]:
This implies,
v(tn;xn) v(0;x) (k0 +p0)tnvinf (k0 +p0)t0vinf:
Letting n!1, we get
vinf (k0 +p0)t0vinf >vinf;
which is a contradiction.
(2) We prove the case that u1(t;x) and u2(t;x) are bounded sub- and super-solutions of
(1.7). Other cases can be proved similarly.
Let u1(t;x) and u2(t;x) be bounded sub-solution and super-solution of (1.7) respectively.
De ne v(t;x) = e t(u2(t;x) u1(t;x)) and
p = 1 +f1(x;u2(t;x)) + [u1(t;x):
Z 1
0
@f1
@u (x;su
1(t;x) + (1 s)u2(t;x))ds]v(t;x)
for t2[0;T): Then v satis es,
@v
@t 1
Z
D
(y x)v(t;y)dy +p(t;x)v(t;x); x2D:
By the boundedness of u1 and u2, there is > 0 such that inft2[0;T);x2D p(t;x) > 0: Proof
of (2) then follows from the arguments in (1) with p(x) and p0(x) being replaced by p(t;x)
and sup(t;x)2[0;T) Dp(t;x) respectively.
(3) We prove the case that i = 1. Other cases can be proved similarly.
There is L > 0 such that u0(x) L and f1(t;x;L) < 0 for x2D: Let uL(t;x) L
for x2D and t2R: Then uL is a super solution of (1:7) on [0;1): Let I(u0) R be the
24
maximal interval of existence of the solution u1(t;:;s;u0) of (1:7) with u1(s; ;s;u0) = u0( ).
Then by (2); 0 u1(t;x;s;u0) L for x 2 D;t 2 I(u0)\[s;1): It then follows that
[s;1) I(u0); and hence u1(t;x;s;u0) exists for all t s:
Proposition 3.4 (Strong monotonicity). (1) If u1;u2 2Xi, u1 u2 and u1 6= u2, then
i(t;s)u1 i(t;s)u2 for all t>s.
(2) If u1;u2 2Xi, u1 u2 and u1 6= u2, then ui(t; ;s;u1) ui(t; ;s;u2) for every t > s
at which both ui(t; ;s;u1) and ui(t; ;s;u2) exist.
Proof. (1) We prove the case i = 1. The cases i = 2 and i = 3 can be proved analogously.
First we prove 1(t;s)u0 0 if u0 2X1nf0g. We claim that e 1K1tu0 0 for t> 0, where
(K1u)(s;x) = RD (y x)u(s;y)dy.
Note that
e 1K1tu0 = u0 + 1tK1u0 + ( 1tK1)
2
2 u0 +:::
Let x0 2 D be such that u0(x0) > 0: Then there is r > 0; > 0 such that u0(x0) > 0 for
x2B(x0;r) :=fy2Djky x0 k 0
for x2B(x0;r + ): By induction ( 1K1)nu0 > 0 for x2B(x0;r + n );n2N: Therefore,
e 1K1tu0 0 for t> 0: Let m> 1 minx2 D;t2Ra1(t;x): Then,
1(t;s)u0 = e m(t s)e 1K1(t s)u0
+
Z t
s
e m(t )e 1K1(t )(m 1 +a1( ; ))u1( ; ;s;u0)d
e m(t s)e 1K1(t s)u0
0
25
for t > s It then follows that 1(t;s)u0 0 for all t > s. Now let u0 = u2 u1: Then
u0 2X+1 nf0g. Hence 1(t;s)u0 0 for t>s, which implies 1(t;s)u1 1(t;s)u2 for all
t>s.
(2) We prove the case i = 1: Other cases can be proved analogously.
Let v(t;x) = u1(t;x;s;u2) u1(t;x;s;u1) for t s at which both u1(t;x;s;u1) and
u1(t;x;s;u2) exist. Then v(t; ) 0 and v(t;x) satis es
@v
@t = 1
Z
D
(y x)v(t;y)dy 1v(t;x) +f(x;u1(t;x;u2))v(t;x)
+[u1(t;x;u1):
Z 1
0
fu(x;su1(t;x;u1) + (1 s)u1(t;x;u2))ds]v(t;x);
x2 D and t s. Proof of (2) then follows from the arguments similar to those in proof of
(1):
Observe that when considering the spatial spreading and front propagation dynamics of
(2.4), we need to consider (2.4) in X and also need to consider the following nonlocal linear
evolution equation,
@u
@t =
Z
RN
e (y x) (y x)u(t;y)dy u(t;x) +a(t;x)u(t;x); x2RN (3.4)
where 2R, 2SN 1, and a(t; ) 2Xp and a(t + T;x) = a(t;x). Note that if = 0 and
a(t;x) = a0(t;x)(:= f(t;x;0)), (3.4) is the linearization of (2.4) at u 0.
Remark 3.5. In space Xp, (3.4) share the same properties as (3.3).
Throughout the rest of this section, we assume that 2 SN 1 and 2 R are xed,
unless otherwise speci ed.
By the same arguments as in Proposition 3.1, for every u0 2 X, (3.4) has a unique
solution u(t; ;u0; ; ;a)2X with u(0;x;u0; ; ;a) = u0(x). Put
(t; ; ;a)u0 = u(t; ;u0; ; ;a): (3.5)
26
Note that if u0 2 Xp, then (t; ; ;a)u0 2 Xp for t 0. Similarly, (2.4) has a unique
(local) solution u(t;x;u0) with u(0;x;u0) = u0(x) for every u0 2X. Also if u0 2Xp, then
u(t;x;u0)2Xp for t in the existence interval of the solution u(t;x;u0).
A continuous function u(t;x) on [0;T) RN is called a super-solution or sub-solution of
(3.4) if @u@t exists and is continuous on [0;T) RN and satis es
@u
@t
Z
RN
e (y x) k(y x)u(t;y)dy u(t;x) +a(t;x)u(t;x); x2RN
or
@u
@t
Z
RN
e (y x) )k(y x)u(t;y)dy u(t;x) +a(t;x)u(t;x); x2RN
for t2(0;T).
For convenience, we would like to restate some comparison properties of solutions to
(2.4) and (3.4) in the following.
Proposition 3.6 (Comparison principle).
(1) If u1(t;x) and u2(t;x) are sub-solution and super-solution of (3.4) 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) Suppose that u1;u2 2Xp and u1 u2, u1 6= u2. Then (t; ; ;a)u1 (t; ; ;a)u2
for all t> 0.
Proof. It follows from Propositions 3.3 and 3.4.
For given 0, let
X( ) =fu2C(RN;R)j x7!e kxku(x) 2 Xg (3.6)
equipped with the norm kukX( ) = supx2RN e kxkju(x)j.
27
Remark 3.7. For every u0 2 X( ) ( 0),the equation (3.4) has a unique solution
u(t; ;u0; ; ) 2 X( ) with u(0;x;u0; ; ) = u0(x). Moreover, Proposition 3.6 holds for
such solutions of (3.4).
Proposition 3.8 (Comparison principle).
(1) If u1(t;x) and u2(t;x) are bounded sub- and super-solutions of (2.4) on [0;T), respec-
tively, and u1(0; ) u2(0; ), then u1(t; ) u2(t; ) for t2[0;T).
(2) If u1;u2 2Xp with u1 u2 and u1 6= u2, then u(t; ;u1) u(t; ;u2) for every t> 0 at
which both u(t; ;u1) and u(t; ;u2) exist.
(3) For every u0 2X+, u(t;x;u0) exists for all t 0.
Proof. If follows from the arguments in Propositions 3.3 and 3.4.
Remark 3.9. Let
~X =fu : RN !Rju is Lebesgue measurable and boundedg
equipped with the norm kuk= supx2RNju(x)j, and
~X+ =fu2 ~Xju(x) 0 8x2RNg:
By general semigroup theory, for any u0 2 X, (2.4) has also a unique (local) solution
u(t; ;u0) 2 ~X with u(0;x;u0) = u0(x). Similarly, we can de ne measurable sub- and
super-solutions of (2.4). Proposition 3.8 (1) and (3) also hold for bounded measurable sub-,
super-solutions and solutions.
28
3.2 Basic properties of principal eigenvalues and principal spectrum points of
nonlocal dispersal operators
Let Ki :Xi!Xi and Hi :D(Hi) Xi!Xi be de ned as follows,
(K1u)(s;x) = (K2u)(s;x) =
Z
D
(y x)u(s;y)dy;
(K3u)(s;x) =
Z
RN
(y x)u(s;y)dy;
(H1(a1)u)(s;x) = us 1u(s;x) +a1(s;x)u(s;x);
(H2(a2)u)(s;x) = us 2
Z
D
(y x)dyu(s;x) +a2(s;x)u(s;x);
and
(H3(a3)u)(s;x) = us 3u(s;x) +a3(s;x)u(s;x):
Then,
Li(ai)u = ( iKi +Hi(ai))u; i = 1;2;3:
We denote I as an identity map from Xi to Xi and may write Iu as u and I Hi(ai)
as Hi(ai), etc.. If no confusion occurs, we may write Li(ai) and Hi(ai) as Li and Hi,
respectively.
Observe that if 2R is such that ( Hi) 1 exists, then
( iKi +Hi)u = u
has nontrivial solutions in Xi is equivalent to
iKi( Hi) 1v = v
29
has nontrivial solutions in Xi. Moreover, it can be claimed that is an eigenvalue of Li(ai)
if and only if 1 is an eigenvalue of iKi( Hi) 1.
In fact, if is an eigenvalue of Li(ai), then there exists nonzero v such that Li(ai)v =
( iKi + Hi(ai))v = v. There exists nonzero u such that v = ( Hi) 1u. This implies,
iKi( Hi) 1u = u, showing that 1 is an eigenvalue of iKi( Hi) 1. Conversely, if 1 is an
eigenvalue of iKi( Hi) 1, then there exists nonzero w such that iKi( Hi) 1w = w. Let
v = ( Hi) 1w. Then ( iKi)v = ( Hi)v, which implies Li(ai)v = ( iKi+Hi(ai))v = v,
showing that is an eigenvalue of Li(ai).
Lemma 3.10. Let fung be any bounded sequence in X1. Then for any > maxx2 D(b1(x) +
^a1(x)), Rt 1exp(Rts( 1 +a1( ;y) )un( ;y)d )ds is bounded.
Proof. First of all, it is clear that for any y 2 D and t2R, Rt 1exp(Rts( 1 + a1( ;y)
)un( ;y)d )ds exists. Suppose that kunk M for all n 1. Then
j
Z t
1
exp(
Z t
s
( 1 +a1( ;y) )un( ;y)d )dsj M
Z t
1
exp(
Z t
s
( 1 +a1( ;y) )d )ds
To prove the boundedness of Rt 1exp(Rts( 1 + a1( ;y) )un( ;y)d )ds, let f(t;y) =
Rt
1exp(
Rt
s( 1 +a1( ;y) )d )ds. Then,
f(t+T;y) =
Z t+T
1
exp(
Z t+T
s
( 1 +a1( ;y) )d )ds
=
Z t
1
exp(
Z t+T
s+T
( 1 +a1( ;y) )d )ds
=
Z t
1
exp(
Z t
s
( 1 +a1( ;y) )d )ds
= f(t;y):
Note that f(t;y) is continuous and being the continuous periodic function, it is bounded.
This implies that Rt 1exp(Rts( 1 +a1( ;y) )un( ;y)d )ds is bounded.
30
Proposition 3.11. Let 1 i 3 be given. Hi generates a positive semigroup of contractions
on Xi and for any > maxx2 D(bi(x) + ^ai(x)), iKi( Hi) 1 is a compact operator on Xi.
Proof. We will prove that H1 generates a positive semigroup 1(s) of contraction onX1: The
remaining cases can be proved similarly. De ne 1(s) :X1 !X1 by
( 1(s)u)(t;x) = e
Rt
t sh1( ;x)d u(t s;x)
where h1(t;x) = a1(t;x) 1: Then, we claim the following
Claim 1: 1(s1 +s2) = 1(s1) 1(s2):
Claim 2: 1(0) = I:
Proof of claim 1: Note that,
( 1(s1) 1(s2)u)(t;x)
= 1(s1)e
Rt
t s2 h1( ;x)d u(t s2;x)
= e
Rt
t s1 h1( ;x)d w(t s1;x)[where w(t;x) = e
Rt
t s2 h1( ;x)d u(t s2;x)]
= e
Rt
t s1 h1( ;x)d e
Rt s1
t s1 s2 h1( ;x)d u(t s1 s2;x)
= e
Rt
t s1 s2 h1( ;x)d u(t s1 s2;x)
= ( 1(s1 +s2)u)(t;x):
Proof of claim 2: Note that,
( 1(0)u)(t;x) = e
Rt
t h1( ;x)d u(t;x)
= u(t;x):
Now, let U(s;t;x;u) be solution of
Us = Ut +h1(t;x)U
31
with initial condition U(0;t;x;u) = u(t;x):
Then by direct computation
U(s;t;x;u) = ( 1(s)u)(t;x)
and
fu2X1j lim
s!0+
1(s)u u
s existsg= D(H1):
Also, from the de nition of 1(s); positivity is obvious. Moreover, k 1(s)uk kuk;(s 0).
Thus, H1 generates positive semigroup of contraction 1(s) on X1:
Next, we prove that for any >maxx2 D(b1(x) + ^a1(x)); 1K1( H1) 1 is a compact
operator on X1: The other cases (i = 2;i = 3) can be proved analogously.
Note that,
1K1( H1) 1u(t;x)
= 1
Z
D
(y x)( H1) 1u(t;y)dy
= 1
Z
D
f (y x)
Z t
1
exp(
Z t
s
( 1 +a1( ;y) )u( ;y)d )dsgdy:
To show the compactness, let fung be any bounded sequence in X1 and let vn = 1K1(
H1) 1un: By the smoothness property of (y x) and Lemma 3.10,
jvn(t;x1) vn(t;x2)j
= j 1K1( H1) 1un(t;x1) 1K1( H1) 1un(t;x2)j
= 1j
Z
D
[ (y x1) (y x2)]( H1) 1un(t;y)dyj
1
Z
D
jf (y x1) (y x2)gj
Z t
1
exp(
Z t
s
( 1 +a1( ;y) )un( ;y)d )ds]dy
M(x2 x1):
32
Then for every > 0 there is > 0 such that if jx1 x2j< , jvn(t;x1) vn(t;x2)j< :
Clearly, for every > 0, there is also > 0 such that ifjt1 t2j< , thenjvn(t2;x) vn(t1;x)j<
. Therefore, fvng is equicontinuous. The compactness of iKi( Hi) 1 then follows by
using Arzela Ascoli theorem.
Put
i(T;ai) = i(T;0); i = 1;2;3;
and let r( i(T;ai)) be the spectral radius of i(T;ai).
Proposition 3.12. For give 1 i 3,
lnr( i(T;ai))
T = lim supt s!1
lnk i(t;s)k
t s :
Proof. First, by ( i(T;ai))n = i(nT;0). it is clear that
lnr( i(T;ai))
T =
ln
n
limn!1
k( i(T;ai))nk
1=no
T lim supt s!1
lnk i(t;s)k
t s :
Next, for any > 0, there is K 1 such that
k( i(T;ai))nk=k i(nT;0)k (r( i(T;ai)) + )n 8n K:
Note that there is M > 0 such that
k i(t;s)k M 8t>s; t s< 1:
For any s < t with t s (K + 2)T, let n1;n2 2Z be such that 0 s n1T < T and
0 t n2T i;max, ( Hi) 1
exists. Moreover,
( Hi) 1v
(t;x) M
i(x)
v(x)
for any i;max < i;max + 1 and any v2X+i with v(t;x) v(x), where
M = inf
s t s+T;s;t2R
exp(
Z t
s
(min
x2Di
(bi(x) +ai( ;x)) i;max 1)d ):
Proof. First of all, by Floquet theory for periodic ordinary di erential equations, for any
2C with Re > i;max, ( Hi) 1 exists. Moreover, for any v2Xi iXi, we have
( Hi) 1v
(t;x) =
Z t
1
exp (
Z t
s
(bi(x) +ai( ;x) )v( ;x)d )ds:
Hence for any v2Xi with v(t;x) v(x), we have
( Hi) 1v
(t;x) =
nZ t
1
exp (
Z t
s
(bi(x) +ai( ;x) )d )ds
o
v(x):
If i;max < i;max + 1, then
Z t
1
exp (
Z t
s
(bi(x) +ai( ;x) )d )ds M
i(x)
;
where
M = inf
s t s+T;s;t2R
exp(
Z t
s
(min
x2Di
(bi(x) +ai( ;x)) i;max 1)d )
36
(see the arguments of [28, Lemma 3.6]). It then follows that for any i;max < i;max + 1
and v2X+i with v(t;x) v(x),
( Hi) 1v
(t;x) M
i(x)
v(x):
The proposition is thus proved.
Proposition 3.15. For given 1 i 3, s(Li;ai) > maxx2 D i(x) i there is >s(Li;ai)
such that r( iKi( Hi) 1) > 1.
Proof. By Propositions 3:13,
i;max = sup (Hi):
By Proposition 3:11, iKi( Hi) 1 is a compact operator for any 2C with Re > i;max.
It then follows from [6, Theorem 2.2] that s(Li;ai) > i;max i there is > i;max such that
r( iKi( Hi) 1) > 1.
Proposition 3.16. For given 1 i 3, if there is 0 > maxx2 Di i(x) such that r( iKi( 0
H) 1) > 1, then there is i > 0(> maxx2 D i(x)) such that r( iKi( i H) 1) = 1 and i
is an isolated eigenvalue of iKi +Hi of nite multiplicity with a positive eigenfunction.
Proof. Suppose that there is 0 > i;max such that r( iKi( 0 H) 1) > 1. Then by Propo-
sition 3:15, s(Li;ai) > i;max. Moreover, by [6, Theorem 2.2], r( iKi(s(Li;ai) H) 1) = 1,
and s(Li;ai) is an isolated eigenvalue of iKi + Hi of nite multiplicity with a positive
eigenfunction.
Proposition 3.17. For given 1 i 3, if 2R is an eigenvalue of Li(ai) with a positive
eigenfunction, then it is geometric simple.
37
Proof. Suppose that (t;x) is a positive eigenfunction of Li associated with . By Proposi-
tion 3.3, (t;x) > 0 for t2R and x2 Di. Assume that (t;x) is also an eigenfunction of
Li associated with . Then there is a2R such that w(t;x) = (t;x) a (t;x) satis es
w(t;x) 0 8t2R; x2 Di and w(t0;x0) = 0
for some t0 2 R and x0 2 Di. By Proposition 3.3 again, w(t;x) 0 and then (t;x) =
a (t;x). This implies that is geometric simple.
Proposition 3.18. For 1 i 3, s(Li;ai) = lnr( i(T;ai))T .
Proof.
s(Li;ai) = lim sup
t s!1
lnk i(t;s;ai)k
t s :
By Proposition 3.12,
lim sup
t s!1
lnk i(t;s;ai)k
t s =
lnr( i(T;ai))
T :
The proposition thus follows.
Proposition 3.19. For 1 i 3, if ani 2 Xi and ani ! ai in Xi as n ! 1, then
s(L1;ani )!s(Li;ai) as n!1.
Proof. We prove the case i = 1: The remaining cases can be proved similarly.
By Propositions 3:18, s(L1;a1) = lim supt s!1 lnk 1(t;s)kt s .
First, for given a11 and a21 with a11 a21, let i(t;s);i = 1;2, be the evolution operators
generated by (3:1) with a1(t;x) replaced by ai1(t;x);i = 1;2 respectively. We claim that
k 1(t;s)k k 2(t;s)k:
In fact, for any given u0 2X1 with u0 0; by Proposition 3:3 , i(t;s)u0 0 for i = 1;2
and s t: Assume, v(t;x) = 2(t;s)u0 1(t;s)u0: Then v satis es,
38
vt = 1
Z
D
(y x)v(t;y)dy 1v(t;x) +a21(t;x)v(t;x) + (a21 a11) 1(t;s)u0
1
Z
D
(y x)v(t;y)dy 1v(t;x) +a21(t;x)v(t;x):
By Proposition 3:3, v(t;x) 0 and claim is thus proved.
Next, let (t;s) be the evolution operators generated by (3:1) with a1(t;x) being
replaced by a1(t;x) : Then we have (t;s) = e (t s) (t;s): Therefore,
s(L1;a1 ) = s(L1;a1) :
By the rst and next arguments it follows that s(L1;an1 ) !s(L1;a1) as n!1 whenever
an1 !a1 as n!1.
39
Chapter 4
Principal Eigenvalue and Principal Spectrum Point Theory
This chapter contains two sections. In the rst section, we investigate the existence
and lower bounds of principal eigenvalues of nonlocal dispersal operators with time periodic
dependence and prove Theorems A-C. Most results in this section have been published (see
[42]). In the sequel section, we explore some other important properties about principal
spectrum point and principal eigenvalues of nonlocal dispersal operators. Most results in
this section are submitted for publication (see [43]).
4.1 Proofs of Theorems A-C
First of all, we prove an important technical lemma, which will also be used in next
section.
Lemma 4.1. For any ai2Xi and any > 0, there is ai; 2Xi satisfying that
kai ai; kXi < ;
bi + ai; is CN, bi + ^ai; attains its maximum at some point x0 2 Int(Di), and the partial
derivatives of bi + ^ai; up to order N 1 at x0 are zero.
Proof. We prove the case i = 1 or 2. The case i = 3 can be proved similarly (it is simpler).
First, let ~x0 2 Di be such that
i(~x0) = max
x2 D
i(x):
For any > 0, there is ~x 2Int(Di) such that
i(~x0) (~x ) < : (4.1)
40
Let ~ > 0 be such that
B(~x ;~ ) Di;
where B(~x ;~ ) denotes the open ball with center ~x and radius ~ .
Note that there is ~hi 2C( Di) such that 0 ~hi(x) 1, ~hi(~x ) = 1, and supp(~hi)
B(~x ;~ ). Let
~ai; (t;x) = ai(t;x) + ~hi(x)
and
~ i; (x) = bi(x) + ^ai(x) + ~hi(x):
Then ~ai; and ~ i; are continuous on Di,
k~ai; aik (4.2)
and ~ i; attains its maximum in Int(Di).
Let ~Di RN be such that Di ~Di. Note that ~ i; can be continuously extended to ~Di.
Without loss of generality, we may then assume that ~ i; is a continuous function on ~Di and
assume that there is x0 2Int(Di) such that ~ i; (x0) = supx2~Di ~ i; (x). Observe that there is
> 0 and i; 2C( ~Di) such that B(x0; ) Di,
0 i; (x) ~ i; (x) 8 x2 ~Di; (4.3)
and
i; (x) = ~ i; (x0) 8 x2B(x0; ):
Let
(x) =
8
>><
>>:
C exp( 1kxk2 1) if kxk< 1
0 if kxk 1;
41
where C > 0 is such that RRN (x)dx = 1. For given > 0, set
(x) = 1 N (x ):
Let
i; ; (x) =
Z
~Di
(y x) i; (y)dy:
By [15, Theorem 6, Appendix C], i; ; is in C1( ~Di) and when 0 < 1,
j i; ; (x) i; (x)j< 8 x2 Di:
It is not di cult to see that for 0 < 1,
i; ; (x) i; (x0) 8x2B(x0; )
and
i; ; (x) = i; (x0) 8x2B(x0; =2):
Fix 0 < 1, and let
i; (x) = i; ; (x):
Then i; attains its maximum at some x0 2 Int(Di), and the partial derivatives of i; up
to order N 1 at x0 are zero, Let
ai; = ~ai; + i; ~ i; :
Then ai; is CN( Di),
kai ai; k kai ~ai; k+k i; ~ i; k< 2
42
and
bi(x) + ^ai; (x) = i; (x):
Therefore, bi+^ai; attains its maximum at some point x0 2Int(D), and the partial derivatives
of bi + ^ai; up to order N 1 at x0 are zero. The lemma is thus proved.
Next, we recall some results proved in [49] and [52].
Lemma 4.2. If
max
x2 Di
^ai(x) min
x2 Di
^ai(x) < i inf
x2 Di
Z
Di
(y x)dy
in the case i = 1;2 and
max
x2 Di
^ai(x) min
x2 Di
^ai(x) < i
in the case i = 3, then s(Li;^ai) > maxx2 Di i(x) (1 i 3).
Proof. See [49] in the case i = 3 and [52] in the case i = 2;3.
Proof of Theorem A. We prove the case i = 1. The other cases can be proved similarly.
First, we assume that s(L1;a1) is an isolated eigenvalue of L1 with a positive eigen-
function (t;x). Let u(t;x) = es(L1;a1)t (t;x). Then u(t;x) is the solution of (3:1) with
u(0; ) = (0; ) 2X+1 . By Proposition 3:3, we must have (t;x) > 0 for t2R and x2 D.
Then
t(t;x) (t;x) + 1
R
D (y x) (t;y)dy
(t;x) 1 +a1(t;x) = s(L1;a1) 8x2
D; t2R:
This implies that
s(L1;a1) = 1 + ^a1(x) + 1T
Z T
0
R
D (y x) (t;y)dy
(t;x) dt 8x2
D
43
and hence
s(L1;a1) > 1 + max
x2 D
^a1(x):
Conversely, assume that s(L1;a1) > 1 + maxx2 D ^a1(x). By Proposition 3:15, there is
>s(L1;a1) such that r( 1K1( H1) 1) > 1. By Proposition 3:16, s(L1;a1) is the isolated
eigenvalue of L1(a1) of nite multiplicity with a positive eigenfunction. Thus s(L1;a1) is the
principal eigenvalue of L1(a1):
Next, we prove Theorem B(1) and (2).
Proof of Theorem B. (1) We prove the case i = 3. The other cases can be proved similarly.
Put
(K u)(t;x) =
Z
D3
1
N ~ (
y x)
)u(t;y)dy:
Assume x0 2 D3 is such that 3(x0) = maxx2 D3 3(x). By Proposition 3.14, for any > 0,
there is M > 0 such that for any > 3(x0) with 3(x0) < and any v 2X+3 with
v(t;x) v(x) and supp(v) fx2 D3j 3(x) < g,
( H3) 1v M
3(x0)
v:
This implies that
3K ( H1)v
Z
D
3M (y x)
3(y) v(y)dy
It then follows from the arguments in [49, Theorem A] that there is 0 > 0 such that for
0 < < 0, s(L3;a3) is the principal eigenvalue of L3(a3).
(2) We prove the case when i = 2. The other cases can be proved similarly. Let x0 2Int(D)
be such that 2(x0) = maxx2 D (x). By Proposition 3.14, there is M > 0 such that
( H2) 1v M
2(x)
v
44
where v(t;x) 1. This implies that
2K2( H2) 1v
Z
D
2M (y x)
2(y) dy:
By the arguments in [49, Theorem B] (see also [52]), for 2(x0) 1,
2K2( H2) 1v>v:
This implies that r( 2K2( H2) 1) > 1. By Propositions 3.16 and 3.17, s(L2;a2) is the
principal eigenvalue of L2(a2).
Before proving Theorem B(3), we rst prove Theorem C.
Proof of Theorem C. We prove the case i = 2. Other cases can be proved similarly.
First of all, if both L2(a2) and L2(^a2) have principal eigenvalues, then by the arguments
in [28, Theorem 4.1],
s(L2;a2) s(L2;^a2):
[For the detailed proof of the last statement, we need a lemma which we will state
without proof. Before stating the lemma, we state the Jensen inequality which will be useful
in proving the lemma. Jensen Inequality: If f is a positive,continuous function de ned on
[0;T] then,
1
T
Z T
0
f(t)dt expf1T
Z T
0
ln[f(t)]dtg
with equality if and only if f is a constant function. Now we state the lemma whose detailed
proof can be found in [28, Theorem 4.1].
Lemma: Let w(x;t) be a positive continuous function de ned on x [0;T] where is com-
pact . Let (x;y) = 1T RT0 w(y;t)w(x;t)dt: Then either w(x;t) is independent of x or there exists
x 2 such that (x ;y) 1 for all y2 with strict inequality for some y:
45
Proof of the last statement: Assume s(L2;a2) = and s(L2;^a2) = . There exists eigen-
functions (t;x) and (x) with (t;x) > 0 for all t and x and (x) > 0 for all x such
that
t + 2
Z
D
(y x) (t;y)dy 2 (t;x) +a2(t;x) (t;x) = (t;x)
and
2
Z
D
(y x) (y)dy 2 (x) + ^a2 (x) = (x);
which implies,
t + 2
Z
D
(y x) (t;y) (t;x)dy 2 +a2 =
and
2
Z
D
(y x) (y) (x)dy 2 (x) + ^a2(x) = :
Integrating the second last equations with respect to t from 0 to T and then mulplying by
1
T we get
= 2
Z
D
(y x) 1T
Z T
0
(t;y)
(t;x)dtdy 2 + ^a2
Now,
= 2
Z
D
(y x)f1T
Z T
0
(t;y)
(t;x)dt
(y)
(x)gdy
= 2
Z
D
(y x) (y) (x)f1T
Z T
0
w(t;y)
w(t;x)dt 1gdy
where w(t;x) = (t;x) (x :
From the lemma mentioned above, the expression withinfgof the above expression is positive
for all y: Since, (y x) and (x) are also nonnegative, it follows that . ]
46
In general, s(L2;a2) (resp. s(L2;^a2)) may not be the principal eigenvalue of L2(a2)
(resp. L2(^a2)). By Lemma 4.1, for any > 0, there is a2; 2X2 such that
ka2; a2kX2 <
s(L2;a2; ) and s(L2;^a2; ) are principal eigenvalues of L2(a2; ) and L2(^a2; ), respectively. By
the above arguments,
s(L2;a2; ) s(L2;^a2; ):
Clearly,
s(L2;a2) s(L2;a2; ) ; s(L2;^a2) s(L2;^a2; ) + :
It then follows that
s(L2;a2) s(L2;^a2) 2
for any > 0 and hence
s(L2;a2) s(L2;^a2):
Finally, we prove Theorem B(3).
Theorem B(3). By Lemma 4.2, s(Li;^ai) is the principal eigenvalue of Li(^ai). By Theorem
A,
s(Li;^ai) > max
x2 Di
i(x):
By Theorem C,
s(Li;ai) > max
x2 Di
i(x):
By Theorem A again, s(Li;ai) is the principal eigenvalue of Li(ai).
47
4.2 Other important properties
In this section, we present some other properties of principal spectrum points and prin-
cipal eigenvalues for time periodic nonlocal dispersal operators. Throughout this section,
r(A) denotes the spectral radius of an operator A on some Banach space.
Let Xp be as in (2.5). Consider the following eigenvalue problem
vt + (K ; I +a( ; )I)v = v; v2Xp; (4.4)
where 2SN 1, 2R, and a( ; )2Xp. The operator a( ; )I has the same meaning as in
(2.8) with a0( ; ) being replaced by a( ; ), and K ; : Xp!Xp is de ned by
(K ; v)(t;x) =
Z
RN
e (y x) (y x)v(t;y)dy: (4.5)
We point out the following relation between (2.4) and (4.4): if u(t;x) = e (x t) (t;x)
with 2Xpnf0g is a solution of the linearization of (2.4) at u = 0,
@u
@t =
Z
RN
(y x)u(t;y)dy u(t;x) +a0(t;x)u(t;x); x2RN; (4.6)
where a0(t;x) = f(t;x;0), then is an eigenvalue of (4.4) with a(t; ) = a0(t; ) or @t +
K ; I +a0( ; )I and v = (t;x) is a corresponding eigenfunction.
Let ( @t +K ; I +a( ; )I) be the spectrum of @t +K ; I +a( ; )I onXp. Let
0( ; ;a) := supfRe j 2 ( @t +K ; I +a( ; )I)g:
Observe that if = 0, (4.4) is independent of and hence we put
0(a) := 0( ;0;a) 8 2SN 1: (4.7)
48
Observe that @t+K ; I+a( ; )I may not have a principal eigenvalue (see an example
in [49]). Recall that
^a(x) = 1T
Z T
0
a(t;x)dt:
The following proposition provides necessary and su cient condition for @t +K ;
I +a( ; )I to have a principal eigenvalue.
Proposition 4.3. 0( ; ;a) is the principal eigenvalue of @t +K ; I + a( ; )I if and
only if 0( ; ;a) > 1 + maxx2RN ^a(x).
Proof. It follows from Theorem A.
The following proposition provides a very useful su cient condition for 0( ; ;a) to be
the principal eigenvalue of @t +K ; I +a( ; )I.
Proposition 4.4. If a(t; ) is CN and the partial derivatives of ^a(x) up to order N 1 at
some x0 are zero (we refer this to as a vanishing condition), where x0 is such that ^a(x0) =
maxx2RN ^a(x), then 0( ; ;a) is the principal eigenvalue of @t +K ; I + a( ; )I for all
2SN 1 and 2R.
Proof. It follows from the arguments of Theorem B(2).
Proposition 4.5. Each 2 ( @t +K ; I + a( ; )I) with Re > 1 + maxx2RN ^a(x) is
an isolated eigenvalue with nite algebraic multiplicity.
Proof. It follows from [6, Proposition 2.1(ii)].
The following theorem shows that the principal eigenvalue of @t +K ; I+a( ; )I (if
it exists) is algebraically simple, which plays an important role in the proof of the existence
of spreading speeds of (2.4).
Theorem 4.6. Suppose that 0( ; ;a) is the principal eigenvalue of @t+K ; I+a( ; )I.
Then 0( ; ;a) is isolated and algebraically simple with a positive eigenfunction ( ; ; ; ),
k ( ; ; ; )k= 1, and 0( ; ;a) and ( ; ; ; ) are smooth in and .
49
Proof. First of all, note that for > 1 + maxx2RN ^a(x), ( I + @t + I aI) 1 exists (see
[42, Proposition 3.5]). For given > 1 + maxx2RN ^a(x), let
(U ; ; u)(t;x) =
Z
RN
e (y x) (y x)( +@t +I aI) 1u(t;y)dy
and
r( ) = r(U ; ; ):
By [42, Proposition 3.6], U ; ; :Xp!Xp is a positive and compact operator.
Next, by [42, Proposition 3.9], 0( ; ;a) is an isolated geometrically simple eigenvalue
of @t +K ; I + a( ; )I. Let 0 = 0( ; ;a). This implies that r( 0) = 1 and r( 0)
is an isolated geometrically simple eigenvalue of U 0; ; with ( ; ; ; ) being a positive
eigenfunction. We claim that r( 0) is an algebraically simple isolated eigenvalue of U 0; ;
with a positive eigenfunction ( ; ), or equivalently, (I U 0; ; )2 = 0 ( 2Xp) i 2
spanf g. In fact, suppose that 2Xpnf0g is such that (I U 0; ; )2 = 0. Then
(I U 0; ; ) = ; (4.8)
for some 2R. We prove that = 0. Assume that 6= 0. Without loss of generality, we
assume that > 0. By (4.8) and U 0; ; = , we have
= U 0; ; + = U 0; ; ( + ): (4.9)
Then by (4.9) and U 0; ; = , we have
+ = U 0; ; ( + ) + ;= U 0; ; ( + 2 )
50
and hence
= U 0; ; ( + ) = U 0; ;
U 0; ; ( + 2 )
= U2 0; ; ( + 2 ):
By induction, we have
= Un 0; ; ( +n ); 8n 1:
This implies that
n = U
n
0; ; (
n + ):
Note that (t;x) > 0 and then
(t;x)
n + (t;x) > 0; 8n 1:
By the positivity of U 0; ; , we then have
(t;x)
n > 0; 8n 1
and then
(t;x)
n (t;x) = (U
n
0; ; (
n))(t;x) > 0; 8n 1:
It then follows that
(t;x) 0
and so
0;
whis is a contradiction. Therefore, = 0 and hence by (4.8),
2spanf g:
51
The claim is thus proved.
Now, we prove that 0( ; ;a) is an algebraically simple eigenvalue of @t +K ; I +
a( ; )I or equivalently, ( @t +K ; I + a( ; )I 0I)2 = 0 i 2 spanf g. By the
above arguments, there are one dimensional subspaceX1;p = span( ) and one-codimensional
subspace X2;p of Xp such that
Xp =X1;p X2;p;
U 0; ; X1;p =X1;p; U 0; ; X2;p X2;p; (4.10)
and
162 (U 0; ; jX2;p):
Suppose that 2Xp is such that
( @t +K ; I +a( ; )I 0I)2 = 0:
Then there is 2R such that
( @t +K ; I +a( ; )I 0I) = : (4.11)
Let i2Xi;p (i = 1;2) be such that
= ( 0I +@t +I aI) 1 1 + ( 0I +@t +I aI) 1 2:
Then
( @t +K ; I +a( ; )I 0I) = (U 0; ; I) 1 + (U 0; ; I) 2
= (U 0; ; I) 2
= :
52
This together with (4.10) implies that 2X2;p and hence = 0. By (4.11), 2spanf g
and hence 0( ; ;a) is an algebraically simple eigenvalue of @t +K ; I +a( ; )I.
Proposition 4.7. Assume 0( ;0;a) > 0 and 0( ; ;a) is the principal eigenvalue for
> 0. Then there is ( )2(0;1) such that
0( ; ( );a)
( ) = inf >0
0( ; ;a)
: (4.12)
Proof. Note that 0( ; ;a) 0( ; ;amin), and
0( ; ;amin) =
Z
RN
e y (y)dy 1 +amin
with 1 as an eigenfunction. Note also that there is k0 > 0 such that (y) k0 for kyk r02 .
Let mn( ) = k0Ry <0;kyk r0
2
( y )n
n! dy. Then, for > 0
Z
RN
e y (y)dy 1 +amin k0
Z
kyk r02
e y dy 1 +amin
= k0
1X
n=0
Z
kyk r02
( y )n
n! dy 1 +amin
m0 +m2( ) 2 +
1X
n=2
m2n( ) 2n 1 +amin
Let m := inf
2SN 1
m2( )(> 0). We then have 0( ; ;a) m0+m 2 1+amin !1 as !1. By
0( ;0;a) > 0, 0( ; ;a) !1 as ! 0+. This together with the smoothness of 0( ; ;a)
(see Theorem 4.6) implies that there is ( ) such that (4.12) holds.
Proposition 4.8. For given 2SN 1, suppose that 0( ; ;a) is the principal eigenvalue
of @t +K ; I +a( ; )I for all 2R. Then 0( ; ;a) is convex in .
Proof. First, recall that (t; ; ;a) is the solution operator of (3.4). Let
p(T; ; ;a) = (T; ; ;a)jXp:
53
By [42, Proposition 3.10], we have
r( p(T; ; ;a)) = e 0( ; ;a)T:
Note that (t; ;0;a) is independent of 2SN 1. We put
~ (t;a) = (t; ;0;a) (4.13)
for 2SN 1. For given u0 2X and 2R, if we let u ; 0 (x) = e x u0(x), then u ; 0 2X(j j).
By the uniqueness of solutions of (3.4), we have that for given u0 2X, 2SN 1, and 2R,
(t; ; ;a)u0 = e x ~ (t;a)u ; 0 : (4.14)
Next, observe that for each x2RN, there is a measure m(x;y;dy) such that
(~ (T;a)u0)(x) =
Z
RN
u0(y)m(x;y;dy): (4.15)
Moreover, by (~ (T;a)u0( piei))(x) = (~ (T;a)u0( ))(x piei) for x 2 RN 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) (4.16)
for i = 1;2; ;N. By (4.14), we have
( (T; ; ;a)u0)(x) =
Z
RN
e (x y) u0(y)m(x;y;dy); u0 2X:
54
Let ^ 0( i) := r( p(T; ; i)). By the arguments of [49, Theorem A (2)],
ln[^ 0( 1)] [^ 0( 2)]1 ln(r( p(T; ; 1 + (1 ) 2)):
Thus, by r( (T; ; ;a) = e 0( ; ;a)T, we have
0( ; 1;a) + (1 ) 0( ; 2;a) 0( ; 1 + (1 ) 2;a);
that is, 0( ; ;a) is convex in .
For a xed 2SN 1 and a2Xp, we may denote 0( ; ;a) by ( ).
Proposition 4.9. Let 2SN 1 and a2Xp be given. Assume that (4.4) has the principal
eigenvalue ( ) for 2R and that (0) > 0. Then we have:
(i) 0( ) < ( ) for 0 < < ( ):
(ii) For every > 0, there exists some > 0 such that for < < ( ),
0( ) < (
( ))
( ) + :
Proof. It follows from Theorem 4.6, Propositions 4.7, 4.8, and the arguments of [49, Theorem
3.1].
Proposition 4.10. For any > 0 and M > 0, there are a ( ; ) satisfying the vanishing
condition in Proposition 4.4 such that
a(t;x) a (t;x) a(t;x) a+(t;x) a(t;x) +
and
jr( p(T; ; ;a) r( p(T; ; ;a )j<
55
for 2SN 1 and j j M.
Proof. It follows from [42, Lemma 4.1] and the fact that
p(T; ; ;a ) = e T p(T; ; ;a):
56
Chapter 5
Time Periodic Positive Solutions of Nonlocal KPP Equations in Periodic Media
In this chapter, we consider applications of the principal eigenvalue theory established
in the previous section to time periodic KPP equations with nonlocal dispersal. Main results
of this chapter have been published (see [42]).
For given u1;u2 2X++1 (= X++2 ) or u1;u2 2X++3 , we de ne
(u1;u2) = inffln j 1 u1( ) u2( ) u1( ); 1g: (5.1)
Observe that for u1;u2 2X++1 (= X++2 ) or u1;u2 2X++3 , there is 1 such that
(u1;u2) = ln :
Proposition 5.1. Let 1 i 3 be given.
(1) For any u0;v0 2X++i , (ui(t; ; 0;u0);ui(t; ; 0;v0)) decreases as t increases.
(2) For any u0;v0 2X++i , if u0 6= v0, then (ui(t; ; 0;u0);ui(t; ; 0;v0)) strictly decreases
as t increases.
(3) For any 0 > 0, there is 0 > 0 such that for any u0;v0 2X++i satisfying that
inf
0 t T;x2 D
fui(t;x; 0;u0);vi(t;x; 0;v0g 0
and
(u0;v0) 1 + 0;
57
there holds
(ui(T; ; 0;u0);ui(T; ; 0;v0)) (u0;v0) 0:
Proof. We prove the case i = 1. The other cases can be proved similarly.
(1) For any u0;v0 2X++1 , there is 1 such that
1
v0 u0 v0
and
(u0;v0) = ln :
By Proposition 3.3, for any t> 0, we have
u1(t; ; 0;u0) u1(t; ; 0; v0):
Let w(t;x) = u1(t;x; 0;v0). Then w(0;x) = v0(x) and
@tw =
Z
D
(y x)w(t;y)dy w(t;x) +w(t;x)f(t;x;u1(t;x; 0;v0))
=
Z
D
(y x)w(t;y)dy w(t;x) +wf(t;x;w(t;x))
+w[f(t;x;u1(t;x; 0;v0)) f(t;x;w(t;x))]
Z
D
(y x)w(t;y)dy w(t;x) +wf(t;x;w(t;x)):
This together with Proposition 3.3 implies that
w(t;x) = u1(t;x; 0;v0) u1(t;x; 0; v0) u1(t;x; 0;u0):
Similarly, we can prove that
1
u1(t;x; 0;v0) u1(t;x; 0;u0):
58
Therefore
(u1(t; ; 0;u0);u1(t; ; 0;v0)) ln (u0;v0)
for t> 0. Repeating the above arguments, we have
(u1(t; ; 0;u0);u1(t; ; 0;v0)) = (u1(t s; ;s;u(s; ; 0;v0));u1(t s; ;s;u(s; ; 0;v0))
(u1(s; ; 0;u0);u1(s; ; 0;v0))
for any 0 s 1 such that (u0;v0) = ln . As in
(1), let w(t;x) = u1(t;x; 0;v0). Then w(0;x) = v0(x) and
@tw =
Z
D
(y x)w(t;y)dy w(t;x) +w(t;x)f(t;x;u1(t;x; 0;v0))
=
Z
D
(y x)w(t;y)dy w(t;x) +wf(t;x;w(t;x))
+w[f(t;x;u1(t;x; 0;v0)) f(t;x;w(t;x))]
Z
D
(y x)w(t;y)dy w(t;x) +wf(t;x;w(t;x)) + 0
for some 0. This implies that
@tw(0;x) @tu1(0;x; 0; v0) + 0:
Hence
w(t;x) = u1(t;x; 0;v0) u1(t;x; 0; v0) + ~ 0
for some ~ 0 > 0 and 0 0, there is 0 > 0 such that for
any u0;v0 2X++i with inf0 t T;x2 Dfu1(t;x; 0;u0);v1(t;x; 0;v0g 0 and (u0;v0) 1 + 0,
there holds
(u1(T; ; 0;u0);u1(T; ; 0;v0)) (u0;v0) :
Proof of Theorem E. We prove the case when i = 1. Other cases can be proved similarly.
First of all, for given M 1, u(t;x) M is a supersolution of (1.7). This together
with Proposition 3.3 implies that ui(nT;x; 0;M) decreases as t increases. Let
u+(x) = limn!1ui(nT;x; 0;M):
Next, by Lemma 4.1, there are aki 2Xi such that s(Li;aki ) is the principal eigenvalue of
Li(aki ) with
aki (t;x) 0 for k 1.
Fix a k 1 such that s(Li;aki ) > 0. Then u = ki (t;x) is a subsolution of (1.7) for
0 < 1.
This together with Proposition 3.3 implies that u(nT;x; 0; ki (0; )) increases as n in-
creases. Let
u (x) = lim
k!1
u(kT;x; 0; ni ):
We claim that
u (x) u+(x):
In fact, Assume that u (x)6 u+(x). Observe that
ki (0; ) ui(T; ; 0; ki (0; )) ui(2T; ; 0; ki (0; ))
ui(2T; ; 0;M) ui(T; ; 0;M) M
There are n > 1 such that
1 > 2 > 3 >
and
(ui(nT; ; 0; ki (0; ));ui(nT; ; 0;M)) = ln n:
Let
= limn!1 n:
Then
1
u
(x) u+(x) u (x)
61
and we must have > 1. Therefore,
inf
n 1;0 t T;x2 D
fui(t;x; 0; ki (0; ));ui(t;x; 0;M)g> 0
and
inf
n 1
(ui(nT; ; 0; ki (0; ));ui(nT; ; 0;M)) > 0:
By Proposition 5.1 (3), there is 0 > 0 such that
ln n+1 ln n 0
and hence
ln = limn!1ln n = 1;
which is a contradiction, and therefore
u (x) u+(x):
Observe that u+(x) is upper semicontinuous and u (x) is lower semicontinuous.
Hence
u i( ) := u+( )(= u ( ))2X++i
Moreover, by Dini?s Theorem,
limn!1ui(nT;x; 0;M) = u i(x)
uniformly in x2 D. We then have
ui(T; ; 0;u i) = limn!1ui(T; ; 0;ui(nT; ; 0;M)) = limn!1ui((n+ 1)T; ; 0;M) = u i( ):
62
This implies that ui(t;x; 0;u i) is a positive time periodic solution, and the existence of time
periodic positive solutions of ( 1.7) is thus proved.
Now suppose that u1(t;x) and u2(t;x) are two time periodic positive solutions of ( 1.7).
Since (u1(t; );u2(t; )) strictly decreases if u1 6= u2, we must have u1 = u2. This proves the
uniqueness of time periodic positive solutions.
Finally, for any u0 2X+nf0g, ui(t; ;u0)2Int(X+) for t> 0. Take 0 < 1, k 1,
and M 1, we have
ki (0; ) u0( ) M:
Then
ui(t;x; 0; ki (0; )) ui(t;x; 0;u0) u(t;x; 0;M)
for t 0. It then follows that
lim
t!1
(ui(t;x; 0;u0) u i(t;x)) = 0
uniformly in x2 D. Therefore, the unique time periodic positive solution is asymptotically
stable.
63
Chapter 6
Spatial Spreading Speed of Nonlocal KPP Equations in Periodic Media
In this chapter, we investigate the existence and characterization of the spreading speeds
of (2.4) and prove Theorems G and H. The main results of this chapter have been submitted
for publication (see [43]). Throughout this chapter, we assume (H1) and (H2). u(t;x;u0)
denotes the solution of (2.4) with u(0;x;u0) = u0(x). By Theorem E, (2.4) has a unique
positive periodic solution u+( ; )2X+p .
To prove Theorems G and H, we rst prove some lemmas.
Consider the space shifted equations of (2.4),
@u
@t =
Z
RN
(y x)u(t;y)dy u(t;x) +u(t;x)f(t;x+z;u(t;x)); x2RN; (6.1)
where z2RN. Let u(t;x;u0;z) be the solution of (6.1) with u(0;x;u0;z) = u0(x) for u0 2X.
Lemma 6.1. (1) Let 2SN 1, u0 2 ~X+ with lim inf
x ! 1
u0(x) > 0 and lim sup
x !1
u0(x) = 0,
and c2R be given. If there is 0 such that
lim inf
x cnT;n!1
u(nT;x;u0;z) 0 uniformly in z2RN; (6.2)
then for every c0 0 such that
lim inf
kxk cnT0;n!1
u(nT;x;u0;z) 0 uniformly in z2RN; (6.4)
then for every c0 0 and n2N, let un2X( 0 + 1) be such that
un(x) =
(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~ (T)unkX( 0+1) !0 as n!1: This implies
that Z
RN
un(y)m(x;y;dy)!0 as n!1
65
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
uniformly for j j 0 and x in bounded subset of RN. By (4.16), 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.
Without loss of generality, in the rest of this section, we assume that the time period
T = 1.
Lemma 6.3. For given 2SN 1, if 0( ; ;a0) is the principal eigenvalue of @t +K ;
I +a0( ; )I for any > 0, then
c sup( ) inf >0 0( ; ;a0) : (6.5)
Proof. For given 2SN 1, put ( ) = 0( ; ;a0). For any > 0, suppose that ( ; ; )2
X+p and
[ @t + (K ; I +a0( ; )I)] ( ;t;x) = ( ) ( ;t;x):
66
Since f(t;x;u) = f(t;x;0) + fu(t;x; )u for some 0 u, we have, by assumption (H1),
f(t;x;u) f(t;x;0) for u 0. If u0 2X+ , then
u(t;x;u0;z) ( (t; ;0;a0( ; +z))u0)(x) for x;z2RN: (6.6)
It can easily be veri ed that
( (n; ;0;a0( ; +z))~u0)(x) = Me (x n~c) ( ;1;x+z)
= Me (x n~c) ( ;0;x+z)
with ~u0(x) = Me x ( ;0;x + z) for ~c = ( ) and M > 0. For any u0 2X+( ), choose
M > 0 large enough such that ~u0 u0. Then by Propositions 3.6 and 3.8, we have
u(n;x;u0;z) ( (n; ;0;a0( ; +z))u0)(x)
( (n; ;0;a0( ; +z))~u0)(x)
= Me (x n~c) ( ;0;x+z):
Hence,
lim sup
x nc;n!1
u(n;x;u0;z) = 0 for every c> ~c
uniformly in z2R. This together with Lemma 6.1 implies that c sup( ) ( ) for any > 0
and hence (6.5) holds.
Lemma 6.4. For given 2SN 1, if 0( ; ;a0) is the principal eigenvalue of @t +K ;
I +a0( ; )I for any > 0, then
c inf( ) inf >0 0( ; ;a0) : (6.7)
Proof. We prove (6.7) by modifying the arguments in [35] and [55].
67
Observe that, for every 0 > 0, there is b0 > 0 such that
f(t;x;u) f(t;x;0) 0 for 0 u b0; x2RN: (6.8)
Hence if u0 2X+ 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;a0( ; +z))u0)(x) (6.9)
for x2RN and z2RN.
Let r( ) be the spectral radius of (1; ; ;0). Then ( ) = lnr( ) and r( ) is an eigen-
value of (1; ; ;a0( ; )) with a positive eigenfunction ( ;x) := ( ;1;x)(= ( ;0;x)).
By Proposition 4.9, for any 1 > 0, there is 1 such that
0( ) < (
( ))
( ) + 1 (6.10)
for 1 < < ( ). In the following, we x 2( 1; ( )). By Proposition 4.9 again, we
can choose 0 > 0 so small that
( ) 0( ) 3 0 > 0: (6.11)
Let : R![0;1] be a smooth function satisfying that
(s) =
(1 for jsj 1
0 for jsj 2:
(6.12)
By Theorem 4.6, ( ;x) is smooth in . Let
( ;z) = ( ;z) ( ;z) :
68
For given > 0, B > 0, and z2RN, de ne
( ; ;z;B) = 1 tan 1
R
RN ( ;y)e
(y z) sin ( (y z) + ( ;y)) (ky zk=B)m(z;y;dy)
R
RN ( ;y)e
(y z) cos ( (y z) + ( ;y)) (ky zk=B)m(z;y;dy):
By Lemma 6.2, ( ; ;z;B) is well de ned for any B > 0 and 0 < 1, and
lim !0 ( ; ;z;B) =
R
RN ( ;y)e
(y z) ( (y z) + ( ;y)) (ky zk=B)m(z;y;dy)
R
RN ( ;y)e
(y z) (ky zk=B)m(z;y;dy)
uniformly in z2RN and B > 0. By Lemma 6.2 again,
lim
B!1
Z
RN
( ;y)e (y z) (ky zk=B)m(z;y;dy) = r( ) ( ;z) (6.13)
uniformly in z2RN and
limB!1
hR
RN ( ;y)e
(y z) ((y z) ) (ky zk=B)m(z;y;dy)
+RRN ( ;y)e (y z) (ky zk=B)m(z;y;dy)
i
= RRN ( ;y)e (y z) ( (y z) )m(z;y;dy) +RRN ( ;y)e (y z) m(z;y;dy)
= r0( ) ( ;z) +r( ) ( ;z) (6.14)
uniformly in z2RN.
By (6.13) and (6.14), we can choose B 1 and x it so that
Z
RN
( ;y)e (y z) (ky zk=B)m(z;y;dy) e ( ) 0 ( ;z); z2RN; (6.15)
(B +j ( ; ;z;B)j+j ( ;z)j) < ; z2RN; 0 < 1;
( ;z) + ( ; ;z;B) < 0( ) 0 ; z2RN; 0 < 1; (6.16)
and
( ;z) ( ; ;z;B) < 0( ) + 1; z2RN; 0 < 1: (6.17)
69
For given 2 > 0 and > 0, de ne
v(s;z) =
(
2 ( ;z)e s sin (s ( ;z)); 0 s ( ;z)
0; otherwise:
(6.18)
Let
v (x;s;z) = v(x +s ( ;z) + ( ; ;z;B);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;B) = ( ;z) + ( ; ;z;B):
Then for 0 s ( ;z) , we have
u(1;0;v ( ;s;z);z)
e 0 (1; ;0;a0( ; +z))v ( ;s;z)
2e 0RRN
h
( ;y)e [(y z) +s+ ( ; ;z:B)] sin [(y z) +s+ ( ; ;z;B) ( ;y)]
(ky zk=B)
i
m(z;y;dy)
= e 0v(s;z)e ( ; ;z;B) sec ( ; ;z;B) ( ;z) RRN
h
( ;y)e (y z) cos ( (y z) + ( ;y))
(ky zk=B)
i
m(z;y;dy):
Observe that
lim !0e 0e ( ; ;z;B) sec ( ; ;z;B) ( ;z) RRN
h
( ;y)e (y z) cos ( (y z) + ( ;y))
(ky zk=B)
i
m(z;y;dy)
e 0e 0( ) 0e ( ) 0 by (6:15); (6:16)
70
= e ( ) 0( ) 3 0
> 1 (by (6:11)):
It then follows that for 0 s ( ;z) ,
u(1;0;v ( ;s;z);z) v(s;z) = v (( ( ;z) ( ; ;z;B)) ;s;( k( ;z)+ ( ; ;z;B)) +z):
Clearly, the above equality holds for all s 2 R (since v(s;z) = 0 for s ( ;z) or s
( ;z) + ).
Let s(x) be such that v( s(x);x) = maxs2Rv(s;x). Let
v(s;x) =
(v( s(x);x); s s(x)
v(s+ ;x); s s(x) :
Set
v (x;s;z) = v(x +s ( ;z) + ( ; ;z;B);x+z):
We then have
u(1;0; v ( ;s;z);z) v(s;z) = v (( ( ;z) ( ; ;z;B)) ;s;( ( ;z) + ( ; ;z;B)) +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;B);x+z)
71
v(x 0( ) + 1;x+z) (by (6:17))
v
x (
( ))
( ) + 2 1;x+z
(by (6:10))
= v0
x [ (
( ))
( ) 2 1] ;[
( ( ))
( ) 2 1] +z
for z2RN. Let ~c ( ) = ( ( )) ( ) 2 1. Then
u(1;x;v0( ;z);z) v0(x ~c ( ) ;~c ( ) +z)
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 Lemma 6.1 implies that
c inf( ) ~c ( ) = (
( ))
( ) 2 1:
Since 1 is arbitrary, (6.7) holds.
Proof of Theorem G. Fix 2SN 1. Put ( ) = 0( ; ;a0), where a0(t;x) = f(t;x;0). By
Proposition 4.7, there is = ( )2(0;1) such that
inf >0 ( ) = (
)
:
72
It is easy to see that c ( ) exists and c ( ) = ( ) if and only if c inf( ) = c sup( ) = ( ) .
If 0( ; ;a0) is the principal eigenvalue of @t +K ; I + a0( ; )I for all , then by
Lemmas 6.3 and 6.4, we have c ( ) exists and c ( ) = inf >0 ( ) .
In general, let an( ; ) 2 CN(R RN;R)\Xp be such that an satis es the vanishing
condition in Proposition 4.4,
an a0 for n 1 and kan akXp !0 as n!1:
Then,
0( ; ;an)! 0( ; ;a0) as n!1:
Note that for 0 < 1,
uf(t;x;u) u(an(t;x) u) for x2RN; u 0:
By Lemma 6.3 and Proposition 3.8, for any u0 2X+( ) and c> inf >0 0( ; ;an) ,
lim
x ct;t!1
u(t;x;u0) lim
x ct;t!1
un(t;x;u0) = 0;
where un(t;x;u0) is the solution of (6.1) with f(t;x;u) being replaced by fn(t;x;u) =
an(t;x) u. 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) : (6.19)
73
For any > 0, there is 0 > 0 such that
f(t;x;u) f(t;x;0) for t2R; x2RN; 0 __ 0:
Choose M maxt2R;x2RN an(t;x) 0 . By Lemma 6.4 and Proposition 3.8, for any u0 2X+( ) with
supx2RN u0(x) 0,
lim inf
x ct;t!1
u(t;x;u0;z) lim inf
x ct;t!1
un(t;x;u0;z) > 0
for any c< inf >0 0( ; ;an) , where un(t;x;u0;z) is the solution of (6.1) with f(t;x;u) being
replaced by fn(t;x;u) = an(t;x) Mu. This implies that
c inf( ) inf
>0
0( ; ;an)
:
Thus,
c inf( ) inf >0 0( ; ;a0) 2 :
Letting !0, we have
c inf( ) inf >0 0( ; ;a0) : (6.20)
By (6.19) and (6.20),
c sup( ) = c inf( ) = inf >0 0( ; ;a0) :
74
Hence c ( ) exists and
c ( ) = inf >0 0( ; ;a0) :
Proof Theorem H. It can be proved by the arguments similar in [51, Theorem E].
75
Chapter 7
Traveling Wave Solutions of Nonlocal KPP Equations in Periodic Media
In this chapter, we explore the existence and uniqueness of traveling wave solutions of
(2.4) connecting 0 and u+ and prove Theorem I. The main results of this chapter have been
submitted for publication (see [43]). Throughout this chapter, we assume (H1) and (H2).
7.1 Sub- and super-solutions
In this section, we construct some sub- and super-solutions of (2.4) to be used in the
proof of Theorem I. Throughout this subsection, we assume (H1)-(H3) and put a0(t;x) =
f(t;x;0).
For given 2SN 1, let ( ) be such that
c ( ) = 0( ;
( );a0)
( ) :
Fix 2SN 1 and c > c ( ). Let 0 < < 1 < minf2 ; ( )g be such that c = 0( ; ;a0)
and 0( ; ;a0) > 0( ; 1;a0) 1 >c ( ): Put
( ; ) = ( ; ; ); 1( ; ) = ( 1; ; ):
If no confusion occurs, we may write 0( ; ;a0) as ( ).
For given d1 > 0, let
v(t;x;z;d1) = e (x ct) (t;x+z) d1e 1(x ct) 1(t;x+z)
and
u(t;x;z;d1) = maxf0;v(t;x;z;d1)g: (7.1)
76
We may write u(t;x;z) for u(t;x;z;d1) for xed d1 > 0 if no confusion occurs.
Proposition 7.1. For any z2RN, u(t;x;z;d1) is a sub-solution of (6.1) provided that d1
is su ciently large.
Proof. It follows from the similar arguments as in [50, Propsotion 3.2].
For given d2 0, let
v(t;x;z;d2) = e (x ct) (t;x+z) +d2e 1(x ct) 1(t;x+z)
and
u(t;x;z;d2) = minf v(t;x;z;d2);u+(t;x+z)g: (7.2)
We may write v(t;x;z) and u(t;x;z) for v(t;x;z;d2) and u(t;x;z;d2), respectively, if no
confusion occurs.
Proposition 7.2. For any d2 0 and z2RN, u(t;x;z;d2) is a super-solution of (6.1).
Proof. It follows from the similar arguments as in [50, Proposition 3.5].
Proposition 7.3. For u0( ;z)2X+ with u0(x;z) u+(0;x+z), if limx !1 u0(x;z)e x (0;x+z) =
1 uniformly in z2RN and infx O(1);z2RN u0(x;z) > 0, then
lim
x !1
u(t;x+ct ;u0( ;z);z)
e x (t;x+ct +z) = 1 (7.3)
uniformly in t 0 and z2RN, and
inf
x O(1);t 0;z2RN
u(t;x+ct ;u0( ;z);z) > 0: (7.4)
Proof. Assume that u0 2X+ satis es the conditions in the proposition. We rst prove (7.3).
Observe that there are d1;d2 > 0 such that
u(0;x;;z;d1) u0(x;z) u(0;x;z;d2) 8z2RN:
77
By Propositions 7.1 and 7.2,
u(t;x;z;d1) u(t;x;u0( ;z);z) u(t;x;z;d2): (7.5)
This implies that
lim
x !1
u(t;x+ct ;u0( ;z);z)
e x (t;x+ct +z) = 1
uniformly in t 0 and z2RN, i.e., (7.3) holds.
Next we prove (7.4). Without loss of generality, we may assume that u0(x) u0(x piei)
for any ei with ei > 0. By (7.5), there are M 0 such that
u(t;x+ct ;u0( ;z);z) 8t 0; M x M+: (7.6)
Then for any ei with ei > 0,
piei M+ M
and
u0(x) u0(x piei):
Observe that there is ei0 such that ei0 > 0. Then by Proposition 3.8, for any k2N,
u(t;x+ct kpi0ei0;u0;z) = u(t;x+ct ;u0( kpi0ei0);z kpi0ei0)
= u(t;x+ct ;u0( kpi0ei0);z)
u(t;x+ct ;u0( );z):
This together with (7.6) implies that
u(t;x+ct ;u0( ;z);z) 8t 0; M k^pi0 x M+ k^pi0; z2RN; (7.7)
78
where ^pi0 = pi0ei0 (> 0). (7.6) and (7.7) together with ^pi0 0 and M2R such that
u
t+nT;x+cnT +ct ; u(0; ;z cnT ct );z cnT ct
8n 1; x M:
It then follows from Lemma 6.2 that
lim
x ! 1
( (t;x;z) u+(t;x+z) = 0 (7.11)
uniformly in t2R and z2RN.
81
By (7.9), we have
(x;T;z)
= limn!1u
(n+ 1)T;x+c(n+ 1)T ; u(0; ;z c(n+ 1)T );z c(n+ 1)T
= limn!1u
nT;x+cnT ; u(0; ;z cnT );z cnT
= (x;0;z) (7.12)
and
(x;t;z +piei)
= limn!1u
t+nT;x+cnT +ct ; u(0; ;z +piei cnT ct );z +piei cnT ct
= limn!1u
t+nT;x+cnT +ct ; u(0; ;z cnT ct );z cnT ct
= (x;t;z): (7.13)
Moreover, for any x;x02RN with x = x0 ,
(x;t;z x)
= limn!1u
t+nT;x+cnT +ct ; u(0; ;z x cnT ct );z x cnT ct
= limn!1u
t+nT;cnT +ct ; u(0; +x;z x cnT ct );z cnT ct
= limn!1u
t+nT;cnT +ct ; u(0; +x0;z x0 cnT ct );z cnT ct
= limn!1u
t+nT;x0 +cnT +ct ; u(0; ;z x0 cnT ct );z x0 cnT ct
= (x0;t;z x0): (7.14)
By (7.9)-(7.14), (x;t;z) generates a traveling wave solution of (2.4) in the direction of
with speed c.
82
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Appendices
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