Spatial Spread Dynamics of Monostable Equations in Spatially Locally
Inhomogeneous Media with Temporal Periodicity
by
Liang Kong
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 03, 2013
Keywords: Monostable equation, random dispersal, nonlocal dispersal, discrete dispersal,
spreading speed, principal eigenvalue/eigenfunction.
Copyright 2013 by Liang Kong
Approved by
Wenxian Shen, Chair, Professor of Mathematics and Statistics
Georg Hetzer, Professor of Mathematics and Statistics
Xiaoying Han, Associate Professor of Mathematics and Statistics
Bertram Zinner, Associate Professor of Mathematics and Statistics
Abstract
This dissertation is devoted to the study of semilinear dispersal evolution equations of
the form
ut(t;x) = (Au)(t;x) +u(t;x)f(t;x;u(t;x)); x2H;
where H = RN or ZN, A is a random dispersal operator or nonlocal dispersal operator in
the case H= RN and is a discrete dispersal operator in the case H= ZN, and f is periodic
in t, asymptotically periodic in x (i.e. f(t;x;u) f0(t;x;u) converges to 0 as kxk!1 for
some time and space periodic function f0(t;x;u)), and is of KPP type in u.
These type of equations are called as Monostable or KPP type equations, which arise
in modeling the population dynamics of many species which exhibit local, nonlocal and
discrete internal interactions and live in locally spatially inhomogeneous media with temporal
periodicity. The following main results are proved in the dissertation.
Firstly, it is proved that Liouville type property holds for such equations, that is, time
periodic strictly positive solutions are unique. It is proved that if time periodic strictly
positive solutions (if exists) are globally stable with respect to strictly positive perturbations.
Moreover, it is proved that if the trivial solution u = 0 of the limit equation of such an
equation is linearly unstable, then the equation has a time periodic strictly positive solution.
Secondly, spatial spreading speeds of such equations is investigated. It is also proved
that if u 0 is a linearly unstable solution to the time and space periodic limit equation of
such an equation, then the original equation has a spatial spreading speed in every direction.
Moreover, it is proved that the localized spatial inhomogeneity neither slows down nor speeds
up the spatial spreading speeds. In addition, in the time dependent case, various spreading
features of the spreading speeds are obtained.
ii
Finally, the e ects of temporal and spatial variations on the uniform persistence and
spatial spreading speeds of such equations are considered. As in the periodic media case, it
is shown that temporal and spatial variations favor the population?s persistence and do not
reduce the spatial spreading speeds.
iii
Acknowledgments
First and foremost I want to thank my advisor Dr. Wenxian Shen. It has been an honor
to be her Ph.D. student. I will forever be thankful to all her contributions of time, ideas,
and funding to make my Ph.D. experience stimulating and productive. The positivity and
enthusiasm she has for her research was contagious and motivational for me. The patience
and caring she gave inspire me not only to be a better mathematician, but also to be a better
person.
I would like to show my gratitude to my committee members, Drs. Georg Hetzer,
Xiaoying Han and Bertram Zinner, for their time and attention during busy semesters.
I would also like to thank my parents for their unconditional love and support during
the long years of my education. I am truly indebted and thankful to my wife, Li Cheng, for
her sacri ce and support to ensure me stay focus on my graduate study.
In addition, I would like thank the NSF for the nancial support (NSF-DMS-0907752).
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Notations, Hypotheses, De nitions, and Main Results . . . . . . . . . . . . . . 6
2.1 Notions, hypotheses and de nitions . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Comparison principle and global existence . . . . . . . . . . . . . . . . . . . 16
3.2 Convergence on compact subsets and strip type subsets . . . . . . . . . . . . 19
3.3 Part metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Principal eigenvalue theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Principal eigenvalues of time periodic dispersal operators . . . . . . . 26
3.4.2 Principal eigenvalues of spatially periodic dispersal operators . . . . . 28
3.5 Positive solutions and spreading speeds of KPP equations . . . . . . . . . . . 30
3.5.1 Time periodic positive solutions and spreading speeds of KPP equa-
tions in periodic media . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.2 KPP equations in spatially periodic media . . . . . . . . . . . . . . . 32
4 Existence, Uniqueness, and Stability of Time Periodic Strictly Positive Solutions 34
4.1 Uniqueness and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Tail property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Spatial Spreading Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Another Method to Show the Time Independent Case . . . . . . . . . . . . . . . 56
v
6.1 Another method to prove Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . 56
6.2 Spatial Spreading Speeds and Proofs of Theorems 2.2 and 2.3 . . . . . . . . 66
7 E ects of Temporal and Spatial Variations . . . . . . . . . . . . . . . . . . . . . 79
8 Concluding Remarks and Future Plan . . . . . . . . . . . . . . . . . . . . . . . 81
8.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.2.1 Single-species population model . . . . . . . . . . . . . . . . . . . . . 82
8.2.2 Multi-species population model . . . . . . . . . . . . . . . . . . . . . 83
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
vi
Chapter 1
Introduction
In this dissertation, we investigate the Liouville type property and spatial spreading dy-
namics of monostable evolution equations in locally spatially inhomogeneous periodic media,
in particular, we consider the existence, uniqueness, and stability of time periodic positive
solutions and spatial spreading speeds of monostable type dispersal evolution equations in
periodic media with localized spatial inhomogeneity. We also study the in uence of the
inhomogeneity of the underlying media on the spatial spreading speeds of the monostable
equations.
Our model equations are of the form,
ut(t;x) =Au+uf0(t;x;u); x2H; (1.1)
where H= RN or ZN; in the case H= RN,
Au = u
or
(Au)(t;x) =
Z
RN
(y x)u(t;y)dy u(t;x)
( ( ) is a smooth non-negative convolution kernel supported on a ball centered at the origin
and RRN (z)dz = 1), and in the case H= ZN,
(Au)(t;j) =
X
k2K
ak(u(t;j +k) u(t;j))
1
(ak > 0 and K = fk 2Hjkkk = 1g); and f0(t + T;x;u) = f0(t;x + piei;u) = f0(t;x;u)
(T 2R and pi2H are given constants), and @uf0(t;x;u) < 0 for u 0, f0(t;x;u) < 0 for
u 1.
Among others, equation (1.1) is used to model the evolution of population density of
a species. The case that H = RN and Au = u indicates that the environment of the
underlying model problem is not patchy and the internal interaction of the organisms is
random and local (i.e. the organisms move randomly between the adjacent spatial locations,
such A is referred to as a random dispersal operator) (see [1], [2], [9], [21], [23], [24], [45],
[56], [77], [79], [81], [82], [86], etc., for the application in this case). If the environment of
the underlying model problem is not patchy and the internal interaction of the organisms
is nonlocal, (Au)(t;x) = RRN (y x)u(t;y)dy u(t;x) is often adopted (such A is referred
to as a nonlocal dispersal operator) (see [3], [10], [16], [22], [26], [41], etc.). The case that
H = ZN and (Au)(t;j) = Pk2Kak(u(t;j + k) u(t;j)) (which is referred to as a discrete
dispersal operator) arises when modeling the population dynamics of species living in patchy
environments (see [21], [55], [56], [77], [78], [81], [82], [83], etc.). The periodicity of f0(t;x;u)
in t and x re ects the periodicity of the environment. In literature, equation (1.1) is called of
Fisher or KPP type due to the pioneering works of Fisher [24] and Kolmogorov, Petrowsky,
Piscunov [45] on the following special case of (1.1),
ut = uxx +u(1 u); x2R: (1.2)
Central problems about (1.1) include the existence, uniqueness, and stability of time
and space periodic positive solutions and spatial spreading speeds. Such problems have
been extensively studied (see [1]-[8], [11]-[13], [18], [20], [19], [25], [27]-[30], [35], [39], [40],
[44], [49]-[52], [54], [57]-[62], [65]-[76], [80]-[82], [84], etc.). It is known that time and space
periodic positive solutions of (1.1) (if exist) are unique, which is referred to as the Liouville
type property for (1.1). If u 0 is linearly unstable with respect to spatially periodic
2
perturbations, then (1.1) has a unique stable time and space periodic positive solutionu 0(t;x)
and for any 2RN withk k= 1, (1.1) has a spreading speed c 0( ) in the direction of (see
section 2.4 for detail).
In reality, the underlying media of many biological problems is non-periodically inho-
mogeneous. It is therefore of great importance to investigate the dynamics of monostable
evolution equations in various types of non-periodically inhomogeneous media, for example,
in almost periodic media, in periodic media with locally spatial perturbations, etc.. There are
many works on various extensions of the spatial spreading dynamics of monostable evolution
equations in periodic media, see, for example, [4], [31], [37], [59], [68]-[72], etc..
The aim of the current dissertation is to investigate the dynamics of KPP type equations
in periodic media with spatially localized inhomogeneity, in particular, to deal with the
extensions of the above results for (1.1) to KPP type equations in periodic media with
spatially localized inhomogeneity. We hence consider
ut =Au+uf(t;x;u); x2H; (1.3)
where A and H are as in (1.1), @uf(t;x;u) < 0 for u 0, f(t;x;u) < 0 for u 1,
f(t + T;x;u) = f(t;x;u), and jf(t;x;u) f0(t;x;u)j! 0 as kxk!1 uniformly in (t;u)
on bounded sets (f0(t;x;u) is as in (1.1)) (See (H1) in Chapter 2 for detail). We show
that localized inhomogeneity does not destroy the existence and uniqueness of time periodic
positive solutions and it neither slow down nor speed up the spatial spreading speeds. We
also show that temporal and spatial inhomogeneity does not slow down the spatial spreading
speeds. More precisely, we prove
(Liouville type property or uniqueness of time periodic strictly positive solutions) Time
periodic strictly positive solutions of (1.3) (if exist) are unique (see Theorem 2.1(1)).
(Stability of time periodic strictly positive solutions) If (1.3) has a time periodic strictly
positive solution u (t;x), then it is asymptotically stable (see Theorem 2.1(2)).
3
(Existence of time periodic strictly positive solutions) If u = 0 is a linearly unstable solution
of (1.1) with respect to periodic perturbations, then (1.3) has a time periodic strictly positive
solution u (t;x) (see Theorem 2.1(3)).
(Tail property of time periodic strictly positive solutions) If u = 0 is a linearly unstable
solution of (1.1) with respect to periodic perturbations, then u (t;x) u 0(t;x)!0 askxk!
1 uniformly in t (see Theorem 2.1(4)).
(Spatial spreading speeds) If u = 0 is a linearly unstable solution of (1.1) with respect to
periodic perturbations, then for each 2RN with k k = 1, c 0( ) is the spreading speed of
(1.3) in the direction of (see Theorem 2.2).
(E ect of temporal variation) If u = 0 is a linearly unstable solution of
ut(t;x) =Au+u^f0(x;u); x2H; (1.4)
whereH andAis as in (1.1) and ^f0 is the time average of f0(t;x;u) (see (2.19)), then (1.3)
has a time periodic strictly positive solution u (t;x) and for each 2RN with k k= 1, c 0( )
is the spreading speed of (1.3) in the direction of . Moreover,
c 0( ) ^c 0( );
where ^c 0( ) is the spatial spreading speed of (1.4) in the direction of (see Theorem 2.4 (1)).
(E ect of spatial variation) If u = 0 is a linearly unstable solution of
ut(t;x) =Au+u^^f0(u); x2H; (1.5)
where H and A is as in (1.1) and ^^f0(u) is the spatial average of ^f0(x;u) (see (2.21)), then
(1.3) has a time periodic strictly positive solution u (t;x) and for each 2RN with k k= 1,
4
c 0( ) is the spreading speed of (1.3) in the direction of . Moreover,
c 0( ) ^c 0( ) ^^c 0( );
where ^^c 0( ) is the spatial spreading speed of (1.5) in the direction of (see Theorem 2.4 (2)).
We remark that, in the case thatH= RN andA= , Liouvile type property of (1.3) is
discussed in [8] and the methods used in [8] quite rely on the special properties of parabolic
equations. The current thesis recovers the results obtained in [8] by di erent methods, which
apply to all three di erent type dispersal operators.
The rest of the dissertation is organized as follows. In chapter 2, we introduce the
standing notions, hypotheses, and de nitions, and state the main results of the dissertation.
In chapter 3, we present some preliminary materials to be used in the proofs of the main
results. We study the existence, uniqueness, and stability of time periodic positive solutions
of (1.3) in chapter 4. In chapter 5, we explore the spreading speeds of (1.3). We give another
elegant method working on time independent case in chapter 6. In chapter 7, we consider
the temporal and spatial variations on the spatial spreading dynamics of monostable stable
equations. In chapter 8, We will address some remarks and open problems.
5
Chapter 2
Notations, Hypotheses, De nitions, and Main Results
In this chapter, we rst introduce some standing notations, hypotheses, and de nitions.
We then state the main results of the paper.
2.1 Notions, hypotheses and de nitions
In this section, we introduce standing notions, hypotheses, and de nitions. Throughout
this subsection,
H= RN or ZN and pi2H with pi > 0 ( i = 1;2; ;N ) (2.1)
Let
X = Cbunif(H;R) :=fu2C(H;R)ju is uniformly continuous and bounded on Hg (2.2)
with norm kuk= supx2Hju(x)j,
X+ =fu2Xju(x) 0 8x2Hg; (2.3)
and
X++ =fu2X+j inf
x2H
u(x) > 0g: (2.4)
Let
Xp =fu2Xju( +piei) = u( )g; (2.5)
X+p = X+\Xp; (2.6)
6
and
X++p = X++\Xp: (2.7)
For given u;v2X, we de ne
u v (u v) if v u2X+ (u v2X+) (2.8)
and
u v (u v) if v u2X++ (u v2X++): (2.9)
Let Hi and Ai :D(Ai) X!X (i = 1;2;3) be de ned by
H1 = RN; (A1u)(x) = u(x) 8u2D(A1); (2.10)
where D(A1) =fu2Xj@xju( );@2xjxku( )2X; 1 j;k Ng;
H2 = RN; (A2u)(x) =
Z
RN
(y x)u(y)dy u(x) 8u2D(A2) = X; (2.11)
and
H3 = ZN; (A3u)(j) =
X
k2K
ak(u(j +k) u(j)) 8u2D(A3) = X: (2.12)
Let
Xp =fu2C(R H;R)ju(t+T;x+piei) = u(t;x)g (2.13)
7
with norm kuk= maxt2R;x2Hju(t;x)j. For given 2SN 1 and 2R, let A ; :D(A ; )
Xp!Xp be de ned by
(A ; u)(t;x) =
8
>>>>
>>>
>>>>
>>>
<
>>>
>>>
>>>
>>>>
>:
u(t;x) 2 ru(t;x) + 2u(t;x) if H=H1; A=A1
R
RN e
(y x) (y x)u(t;y)dy u(t;x) if H=H2; A=A2
P
k2Kak(e
k u(t;j +k) u(t;j)) if H=H3; A=A3
(2.14)
for u2D(A ; ). Observe that
A ;0 =A 8 2SN 1:
For any given a2Xp, 2SN 1, and 2R, let ( @t+A ; +a( ; )I) be the spectrum
of the operator @t +A ; +a( ; )I :D( @t +A ; +a( ; )I) Xp!Xp,
( @
t +A ; +a( ; )I)u
(t;x) = u
t(t;x) + (A ; u)(t;x) +a(t;x)u(t;x):
Let ; (a) be de ned by
; (a) = supfRe j 2 ( @t +A ; +a( ; )I)g: (2.15)
We call ; (a) the principal spectrum point of @t +A ; + a( ; )I. It equals the principal
eigenvalue of @t +A ; +a( ; )I if it exists (see De nition 3.1 for the de nition of principal
eigenvalue).
It is clear that ;0(a) is independent of 2SN 1 and we may put
(a) = ;0(a):
8
Observe that ( @t+A ; +a( ; )I) is the spectrum of the following eigenvalue problem,
@tu+A ; u+a( ; )Iu = u; u2Xp: (2.16)
When a(t;x) a(x) is independent of t, (2.16) reduces to
A ; u+a( )Iu = u; u2Xp: (2.17)
In the following, to indicate the dependence of X, Xp, Xp on the media, we may put
Xi = X = Cbunif(Hi;R) in the case H=Hi;
Xi;p = Xp =fu2Xiju( +piei) = u( )g in the case H=Hi;
etc..
Consider (1.3). We introduce the following standing hypotheses.
(H0) f(t + T;x;u) = f(t;x;u) for (t;x;u) 2R H R (T > 0 is a given positive num-
ber); f(t;x;u) is C1 in t;u and f(t;x;u), ft(t;x;u), fu(t;x;u) are uniformly continuous in
(t;x;u) 2R H E (E is any bounded subset of R); f(t;x;u) < 0 for all t2R, x2H,
and u M0 (M0 > 0 is some given constant); and inft2R;x2Hfu(t;x;u) < 0 for all u 0.
(H1) f0(t;x;u) satis es (H0), f0(t;x + piei;u) = f0(t;x;u) for (t;x;u) 2R H R, and
jf(t;x;u) f0(t;x;u)j!0 as kxk!1 uniformly in (t;u) on bounded sets.
(H1)0 f0(u) satis es (H0), f(t;x;u) f(x;u), and f(x;u) f0(u)!0 as kxk!1.
(H2) (f0( ; ;0)) > 0.
Observe that f(t;x;u) = 1 u is a typical example which satis es (H0) and (H0) is
referred to as Fisher or KPP type condition. Throughout this section, we assume (H0). By
general semigroup theory (see [32], [64]), for any u0 2X, (1.3) has a unique (local) solution
9
u(t; ;u0) with u(0; ;u0) = u0( ). Furthermore, if f(t;x + piei;u) = f(t;x;u) and u0 2Xp,
then u(t; ;u0)2Xp. To indicate the dependence of u(t;x;u0) on f, we may write u(t;x;u0)
as u(t;x;u0;f).
Observe also that assumption (H1) re ects the localized spatial inhomogeneity of the
media. (H1)0 is a special case of (H1). Assumption (H2) is the linear instability condition
of the trivial solution of (1.3) in the case f(t;x;u) = f0(t;x;u). If f0(t;x;u) f0(u), then
(H2) becomes f0(0) > 0.
Let ^f(x;u), ^f0(x;u), ^^f(u), and ^^f0(u) be de ned as follows.
^f(x;u) = 1
T
Z T
0
f(t;x;u)dt; (2.18)
^f0(x;u) = 1
T
Z T
0
f0(t;x;u)dt; (2.19)
and
^^f(u) = lim
R!1
1
jB(0;R)j
Z
B(0;R)
^f(x;u)dx; (2.20)
^^f
0(u) = limR!1
1
jB(0;R)j
Z
B(0;R)
^f0(x;u)dx; (2.21)
where
B(0;R) =fx2Hjjxij R; i = 1;2; ;Ng
and jB(0;R)j is the Lebesgue measure of B(0;R) in the case H = RN and jB(0;R) is the
cardinality of B(0;R) in the case H= ZN.
Observe that if f(t;x;u) satis es (H0), then so are ^f(x;u) and ^^f(u). If f(t;x;u) and
f0(t;x;u) satisfy (H0) and (H1), then so are ^f(x;u) and ^^f(u). If ^f(x;u) and ^f0(x;u) satisfy
(H0) and (H1), then so are ^^f(u) and ^^f0(u).
Let
SN 1 =f 2RNjk k= 1g: (2.22)
10
For given 2SN 1 and u2X+, we de ne
lim inf
x ! 1
u(x) = lim infr! 1 inf
x2H;x r
u(x):
For given u : [0;1) H!R and c> 0, we de ne
lim inf
x ct;t!1
u(t;x) = lim inft!1 inf
x2H;x ct
u(t;x);
lim sup
x ct;t!1
u(t;x) = lim sup
t!1
sup
x2H;x ct
u(t;x):
The notions lim sup
jx j ct;t!1
u(t;x), lim sup
jx j ct;t!1
u(t;x), lim sup
kxk ct;t!1
u(t;x), and lim sup
kxk ct;t!1
u(t;x) are
de ned similarly. We de ne X+( ) by
X+( ) =fu2X+j lim inf
x ! 1
u(x) > 0; u(x) = 0 for x 1g: (2.23)
De nition 2.1 (Time periodic strictly positive solution). A solution u(t;x) of (1.3) on t2R
is called a time periodic strictly positive solution if u(t + T;x) = u(t;x) for (t;x)2R H
and inf(t;x)2R Hu(t;x) > 0.
De nition 2.2 (Spatial spreading speed). For given 2SN 1, a real number c ( ) is called
the spatial spreading speed of (1.3) in the direction of if for any u0 2X+( ),
lim inf
x ct;t!1
u(t;x;u0) > 0 8cc ( ):
2.2 Main Results
In this section, we state the main results of this dissertation.
The rst theorem is about time periodic strictly positive solutions.
11
Theorem 2.1 (Time periodic strictly positive solutions). Consider (1.3) and assume (H0).
(1) (Liouville type property or uniqueness) If (1.3) has a time periodic strictly positive
solution, then it is unique.
(2) (Stability) Assume that u (t;x) is a time periodic strictly positive solution of (1.3).
Then it is stable and for any u0 2X++, limt!1ku(t; ;u0;f( + ; ; )) u (t+ ; )kXi =
0 uniformly in 2R.
(3) (Existence) Assume also (H1) and (H2). Then (1.3) has a unique time periodic strictly
positive solution u (t;x).
(4) (Tail property) Assume also (H1) and (H2). Then u (t;x) u 0(t;x)!0 as kxk!1
uniformly in t2R, where u (t;x) is as in (3) and u 0(t;x) is the unique time and space
periodic positive solution of (1.1) (see Proposition 3.8 for the existence and uniqueness
of u 0(t;x)).
Remark 2.1. (1) Theorem 2.1 indicates that localized spatial inhomogeneity does not de-
stroy the Liouville type property of (1.3), in particular, it does not destroy the existence
of time periodic positive solution. Moreover, it shows that localized spatial inhomogene-
ity does not a ect the behavior of the time periodic positive solution as the space variable
goes to 1.
(2) Assume (H0) and (H1)0. Then u 0(t;x) is a positive constant, denoted by u0, such that
f0(u0) = 0.
(3) Biologically, Theorem 2.1 implies that if u = 0 is a linearly unstable solution of the
limit equation of (1.3), then the population will persist and is eventually time periodic.
The second theorem is about spatial spreading speeds.
12
Theorem 2.2 (Existence of spreading speeds). Consider (1.3) and assume (H0)-(H2). Then
for any given 2SN 1, (1.3) has a spreading speed c ( ) in the direction of . Moreover,
for any u0 2X+( ),
lim sup
x ct;t!1
ju(t;x;u0) u (t;x)j= 0 8c0
f0(0) + 2
= 2
p
f0(0) in the case H=H1; A=A1;
c 0( ) = inf >0
R
RN e
z (z)dz 1 +f0(0)
in the case H=H2; A=A2
and
c 0( ) = inf
>0
P
k2Kak(e
k 1) +f0(0)
in the case H=H3; A=A3:
For time independent case, we have the following additional result regarding the spread-
ing speeds.
13
Theorem 2.3 (Spreading features of spreading speeds). Assume (H2) and (H1)0 and f0(0) >
0. Then for any given 2SN 1, the following hold.
(1) For each u0 2X+ satisfying that u0(x) = 0 for x2H with jx j 1,
lim sup
jx j ct;t!1
u(t;x;u0) = 0 8c> maxfc 0( );c 0( )g:
(2) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2Hwithjx j r,
lim sup
jx j ct;t!1
ju(t;x;u0) u (x)j= 0 80 sup
2SN 1
c 0( ):
(4) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for kxk r,
lim sup
kxk ct;t!1
ju0(t;x;u0) u (x)j= 0 80 0, then (1.3) has a unique time periodic strictly positive solution u (t;x)
and has a spreading speed c ( ) in the direction of . Moreover, for any u0 2X+( ),
lim sup
x ct;t!1
ju(t;x;u0) u (t;x)j= 0 8c 0, then (1.3) has a unique time periodic strictly positive solution u (t;x)
and has a spreading speed c ( ) in the direction of . Moreover, for any u0 2X+( ),
lim sup
x ct;t!1
ju(t;x;u0) u (t;x)j= 0 8c 0 at which both
u(t; ;u01) and u(t; ;u02) exist. Moreover, if u01 6= u02, then u(t;x;u01) < u(t;x;u02)
for all x2H and t> 0 at which both u(t; ;u01) and u(t; ;u02) exist.
(3) If u01;u02 2X and u01 u02, then u(t; ;u01) u(t; ;u02) for t > 0 at which both
u(t; ;u01) and u(t; ;u02) exist.
Proof. (1) The case that H = H1(= RN) and A = H1(= ) follows from comparison
principle for parabolic equations. We prove the case thatH=H2(= RN) andA=A2. The
case that H=H3(= ZN) and A=A3 can be proved similarly.
Observe that for any t2[0; ),
Z
RN
(y x)u1(t;y)dy =
Z
RNn
(y x)u1(t;y)dy +
Z
(y x)u1(t;y)dy
Z
RNn
(y x)u2(t;y)dy +
Z
(y x)u1(t;y)dy: (3.2)
Let v(t;x) = u2(t;x) u1(t;x). By (3.2),
vt(t;x)
Z
(y x)v(t;y)dy v(t;x) +u2(t;x)f(t;x;u2(t;x)) u1(t;x)f(t;x;u1(t;x))
=
Z
(y x)v(t;y)dy v(t;x) +a(t;x)v(t;x); x2 ; t2(0; );
where
a(t;x) = f(t;x;u2(t;x)) +u1(t;x)
Z 1
0
@uf(t;x;su2(t;x) + (1 s)u1(t;x))ds:
The rest of the proof follows from the arguments of [43, Proposition 2.4].
(2) It follows from (1) with u1(t;x) = u(t;x; ;u01), u2(t;) = u(t;x;u02), and =H.
(3) We provide a proof for the case that H = H2 and A = A2. Other cases can
be proved similarly. Take any > 0 such that both u(t; ;u01) and u(t; ;u02) exist on
[0; ]. It su ces to prove that u(t; ;u02) u(t; ;u01) for t 2 [0; ]. To this end, let
17
w(t;x) = u(t;x;u02) u(t;x;u01). Then w(t;x) satis es the following equation,
wt(t;x) =
Z
RN
(y x)w(t;y)dy w(t;x) +a(t;x)w(t;x);
where
a(t;x) = f(x;u(t;x;u02)) +u(t;x;u01)
Z 1
0
@uf(x;su(t;x;u02) + (1 s)u(t;x;u01))ds:
Let M > 0 be such that M supx2RN;t2[0; ](1 a(t;x)) and ~w(t;x) = eMtw(t;x). Then
~w(t;x) satis es
~wt(t;x) =
Z
RN
(y x) ~w(t;y)dy + [M 1 +a(t;x)] ~w(t;x):
Let K: X!X be de ned by
(Ku)(x) =
Z
RN
(y x)u(y)dy for u2X: (3.3)
Then K generates an analytic semigroup on X and
~w(t; ) = eKt(u02 u01) +
Z t
0
eK(t )(M 1 +a( ; )) ~w( ; )d :
Observe that eKtu0 0 for any u0 2 X+ and t 0 and eKtu0 0 for any u0 2 X++
and t 0. Observe also that u02 u01 2X++2 . By (2), ~w( ; ) 0 and hence (M 1 +
a( ; )) ~w( ; ) 0 for 2 [0;T]. It then follows that ~w(t; ) 0 and then w(t; ) 0 (i.e.
u(t; ;u02) u(t; ;u01)) for t2[0; ].
Proposition 3.2 (Global existence). For any given u0 2X+, u(t; ;u0) exists for all t 0.
18
Proof. Let u0 2X+ be given. There is M 1 such that 0 u0(x) M and f(t;x;M) < 0
for all x2H. Then by Proposition 3.1,
0 u(t; ;u0) M
for any t > 0 at which u(t; ;u0) exists. It is then not di cult to prove that for any > 0
such that u(t; ;u0) exists on (0; ), limt! u(t; ;u0) exists in X. This implies that u(t; ;u0)
exists and u(t; ;u0) 0 for all t 0.
3.2 Convergence on compact subsets and strip type subsets
In this section, we explore the convergence property of solutions of (1.3) on compact
subsets and strip type subsets. As mentioned before, to indicate the dependence of solutions
of (1.3) on the nonlinearity, we may write u(t; ;u0) as u(t; ;u0;f).
Proposition 3.3. Suppose that u0n;u0 2X+ (n = 1;2; ) withfku0nkgbeing bounded, and
fn, gn (n = 1;2 ) satisfy (H0) with fn(t;x;u), gn(t;x;u), and @ufn(t;x;u) being bounded
uniformly in x2H and (t;u) on bounded subsets.
(1) (Convergence on compact subsets) If u0n(x) !u0(x) as n!1 uniformly in x on
bounded sets and fn(t;x;u) gn(t;x;u) as n!1 uniformly in (t;x;u) on bounded
sets, then for each t> 0, u(t;x;u0n;fn) u(t;x;u0;gn)!0 as n!1 uniformly in x
on bounded sets.
(2) (Convergence on strip type subsets) If u0n(x) !u0(x) as n!1 uniformly in x on
any set E with fx jx2Eg being a bounded set of R and fn(t;x;u) gn(t;x;u)!0
as n!1 uniformly in (t;x;u) on any set E with f(t;x ;u)j(t;x;u)2Eg being a
bounded set of R3, then for each t> 0, u(t;x;u0n;fn) u(t;x;u0;gn)!0 as n!1
uniformly in x on any set E with fx jx2Eg being a bounded set of R.
Proof. (1) We prove the case that H=H2 and A2. Other cases can be proved similarly.
19
Let vn(t;x) = u(t;x;u0n;fn) u(t;x;u0;gn). Then vn(t;x) satis es
vnt (t;x) =
Z
RN
(y x)vn(t;y)dy vn(t;x) +an(t;x)vn(t;x) +bn(t;x);
where
an(t;x) =fn(t;x;u(t;x;u0n;fn))
+u(t;x;u0;fn)
Z 1
0
@ufn(t;x;su(t;x;u0n;fn) + (1 s)u(t;x;u0;gn))ds
and
bn(t;x) = u(t;x;u0;gn) fn(t;x;u(t;x;u0;gn)) gn(t;x;u(t;x;u0;gn)) :
Observe thatfan(t;x)gis uniformly bounded and continuous in t and x and bn(t;x)!0 as
n!1 uniformly in (t;x) on bounded sets of [0;1) RN.
Take a > 0. Let
X( ) =fu2C(RN;R)ju( )e k k2Xg
with norm kuk = ku( )e k kk. Note that K : X( ) ! X( ) also generates an analytic
semigroup, where K is as in (3.3), and there are M > 0 and !> 0 such that
ke(K I)tkX( ) Me!t 8t 0;
where I is the identity map on X( ). Hence
vn(t; ) =e(K I)tvn(0; ) +
Z t
0
e(K I)(t )an( ; )vn( ; )d
+
Z t
0
e(K I)(t )bn( ; )d
20
and then
kvn(t; )kX( ) Me!tkvn(0; )kX( ) +M sup
2[0;t];x2RN
jan( ;x)j
Z t
0
e!(t )kvn( ; )kX( )d
+M
Z t
0
e!(t )kbn( ; )kX( )d
Me!tkvn(0; )kX( ) +M sup
2[0;t];x2RN
jan( ;x)j
Z t
0
e!(t )kvn( ; )kX( )d
+ M! sup
2[0;t]
kbn( ; )kX( )e!t:
By Gronwall?s inequality,
kvn(t; )kX( ) e(!+M sup 2[0;t];x2RNjan( ;x)j)t
Mkvn(0; )kX( ) + M! sup
2[0;t]
kbn( ; )kX( )
:
Note that
kvn(0; )kX( ) !0
and
sup
2[0;t]
kbn( ; )kX( ) !0 as n!1:
It then follows that
kvn(t; )kX( ) !0 as n!1
and then
u(t;x;u0n;fn) u(t;x;u0;gn) as n!1
uniformly in x on bounded sets.
(2) It can be proved by similar arguments as in (1) with X( ) being replaced by X ( ),
where
X ( ) =fu2C(H;R)ju ; 2Xg;
with norm kukX ( ) =ku ; kX, where u ; (x) = e jx ju(x).
21
3.3 Part metric
In this section, we investigate the decreasing property of the so called part metric
between two positive solutions of (1.3). Throughout this subsection, we also assume (H0).
First, we introduce the notion of part metric. For given u;v2X++, de ne
(u;v) = inffln j 1 u v u; 1g:
Observe that (u;v) is well de ned and there is 1 such that (u;v) = ln . Moreover,
(u;v) = (v;u)
and
(u;v) = 0 i u v:
In literature, (u;v) is called the part metric between u and v.
Proposition 3.4 (Strict decreasing of part metric). For any > 0, > 0, M > 0, and
> 0 with < M and ln M , there is > 0 such that for any u0;v0 2 X++ with
u0(x) M, v0(x) M for x2H and (u0;v0) , there holds
(u( ; ;u0);u( ; ;v0)) (u0;v0) :
Proof. We give a proof for the case that H = H1 and A = A1. Other cases can be proved
similarly.
Let > 0, > 0, M > 0, and > 0 be given and < M, < ln M . First, note
that by Proposition 3.1, there are 1 > 0 and M1 > 0 such that for any u0 2 X++ with
u0(x) M for x2RN, there holds
1 u(t;x;u0) M1 8t2[0; ]; x2RN: (3.4)
22
Let
1 = 21e (1 e ) sup
t2[0; ];x2RN;u2[ 1;M1M= ]
fu(t;x;u): (3.5)
Then 1 > 0 and there is 0 < 1 such that
1
2 1 0. We prove that de ned in (3.9) satis es the property in the proposition.
For any u0;v0 2 X++ with u0(x) M and v0(x) M for x 2 R and
(u0;v0) , there is 1 such that
1(u0;v0) = ln
and
1
u0 v0
u0:
Note that e M . We rst show that (u(t; ;u0);u(t; ;v0)) is non-increasing in t> 0.
By Proposition 3.1,
u(t; ;v0) u(t; ; u0) for t> 0:
23
Let
v(t;x) = u(t;x;u0):
We then have
vt(t;x) = v(t;x) +v(t;x)f(t;x;u(t;x;u0))
= v(t;x) +v(t;x)f(t;x;v(t;x)) +v(t;x)f(t;x;u(t;x;u0))
v(t;x)f(t;x;v(t;x))
v(t;x) +v(t;x)f(t;x;v(t;x)) 8t> 0:
By Proposition 3.1 again,
u(t; ; u0) u(t; ;u0)
and hence
u(t; ;v0) u(t; ;u0)
for t> 0. Similarly, we can prove that
1
u(t; ;u0) u(t; ;v0)
for t> 0. It then follows that
(u(t; ;u0);u(t; ;v0)) (u0;v0) 8t 0
and then
(u(t2; ;u0);u(t2; ;v0)) (u(t1; ;u0);u(t1; ;v0)) 8 0 t1 t2:
24
Next, we prove that
(u( ; ;u0);u( ; ;v0)) (u0;v0) :
Note that e M and
vt(t;x) = v(t;x) +v(t;x)f(t;x;u(t;x;u0))
= v(t;x) +v(t;x)f(t;x;v(t;x)) +v(t;x)f(t;x;u(t;x;u0))
v(t;x)f(t;x;v(t;x))
v(t;x) +v(t;x)f(t;x;v(t;x)) + 1 80 0 for (t;x) 2R H and ( ; ) 2Xp) and for any 2 ( @t +A ; + a( ; )I),
Re 0.
We remark that ; (a)2 ( @t +A ; +a( ; )I) and if @t +A ; +a( ; )I admits a
principal eigenvalue 0, then 0 = ; (a). We also remark that in the case thatH=Hi and
A = Ai with i = 1 or 3, principal eigenvalue of @t +A ; + a( ; )I always exists. But in
the case thatH=H2 andA=A2, @t +A ; +a( ; )I may not have a principal eigenvalue
(see [17] and [74] for examples).
26
For given a2Xp, let
^a(x) = 1T
Z T
0
a(t;x)dt:
The following proposition is established in [65] regarding principal eigenvalues of time
periodic nonlocal dispersal operators.
Proposition 3.5. (1) If ^a( ) is CN and there is x0 2RN such that ^a(x0) = maxx2RN ^a(x0)
and the partial derivatives of ^a(x) up to order N 1 at x0 are zero, then for any
2SN 1 and 2R, ; (a) is the principal eigenvalue of @t +A ; +a( ; )I.
(2) Let a( ; ) 2Xp be given. For any > 0, there is a ( ; ) 2Xp such that ; (a ) are
principal eigenvalues of @t +A ; +a ( ; )I,
a (t;x) a(t;x) a+(x;) 8(t;x)2R H;
and
; (a+) ; (a) ; (a ) + :
Proof. We only need to prove the case that H=H2 and A=A2.
(1) It follows from [65, Theorem B(1)].
(2) It follows from [65, Proposition 3.10, Lemma 4.1].
The following proposition shows that the temporal variation does not reduce the prin-
cipal spectrum point of dispersal operators.
Proposition 3.6. For any given 2SN 1, 2R, and a2Xp,
; (a) ; (^a):
Proof. It follows from Theorem 6.5 in [73] (see also [42] for the case thatH=H1 andA=A1
and see [43] for the case that H=H2 and A=A2,
27
3.4.2 Principal eigenvalues of spatially periodic dispersal operators
In this subsection, we present some special principal eigenvalue theories for time in-
dependent but spatially periodic dispersal operators with random, nonlocal, and discrete
dispersals.
Recall that, if a(t;x) a(x), the eigenvalue problem (2.16) reduces to the eigenvalue
problem (2.17). To be more precise, when H=H1 and A=A1, (2.16) reduces to
8
>><
>>:
u(x) 2 ru(x) + (a(x) + 2)u(x) = u(x); x2RN
u(x+piei) = u(x); x2RN:
(3.10)
When H=H2 and A=A2, (2.16) reduces to
8>
><
>>:
R
RN e
(y x) (y x)u(y)dy u(x) +a(x)u(x) = u(x); x2RN
u(x+piei) = u(x); x2RN:
(3.11)
When H=H3 and A=A3, (2.16) reduces to
8
>><
>>:
P
k2Kak(e
k u(j +k) u(j)) +a(j)u(j) = u(j); j2ZN
u(j +piei) = u(j); j2ZN:
(3.12)
Observe that when = 0, (3.10), (3.11), and (3.12) are independent of . Observe also
that if u(t;x) = e (x t) (x) is a solution of
ut(t;x) = u(t;x) +a(x)u(t;x); x2RN (3.13)
with ( )2X1;pnf0g, or a solution of
ut(t;x) =
Z
RN
k(y x)u(t;y)dy u(t;x) +a(x)u(t;x); x2RN (3.14)
28
with ( )2X2;pnf0g, or a solution of
ut(t;j) =
X
k2K
ak(u(t;x+j) u(t;j)) +a(j)u(t;j); j2ZN (3.15)
with ( ) 2X3;pnf0g, then is an eigenvalue of (3.10) or (3.11) or (3.12) with ( ) being
a corresponding eigenfunction. If a(x) = f(x;0), then (3.13) (resp. (3.14), (3.15)) is the
linearized equation of (1.3) with f(t;x;u) = f(x;u) and H = H1 and A1 (resp. H = H2
and A=A2, H=H3 and A=A3) at u = 0.
For given ai( )2Xi;p, let ^^ai be the space average of ai( ) (i = 1;2;3), that is,
8
>>>>
>><
>>>>
>>:
^^ai = 1
jDij
R
Diai(x)dx for i = 1;2
^^a3 = 1
#D3
P
j2D3 a3(j);
(3.16)
where
Di = [0;p1] [0;p2] [0;pN]\Hi; i = 1;2;3 (3.17)
and 8
>>>
>>>
<
>>>
>>>
:
jDij= p1 p2 pN for i = 1;2
#D3 = the cardinality of D3:
(3.18)
The following proposition shows a relation between ; (ai) and ; (^^ai) for ai2Xi;p.
Proposition 3.7 (In uence of spatial variation). For given 1 i 3, 2R, 2SN 1,
and ai2Xi;p, there holds
; (ai) ; (^^ai):
Proof. The case i = 1 is well known. The cases i = 2 and 3 follow from [35, Theorem
2.1].
29
We remark that ; (^^ai) (ai2Xi;p, i = 1;2;3) have the following explicit expressions,
8
>>>
>>>>
>>>
>>>>
<
>>>
>>>
>>>
>>>>
>:
; (^^a1) = ^^a1 + 2
; (^^a2) = RRN e z (z)dz 1 + ^^a2
; (^^a3) = Pk2Kak(e k 1) + ^^a3:
(3.19)
3.5 Positive solutions and spreading speeds of KPP equations
In this section, we recall some existing results on the existence, uniqueness, and stability
of time and space periodic positive solutions and spatial spreading speeds of (1.1).
3.5.1 Time periodic positive solutions and spreading speeds of KPP equations
in periodic media
A solution u(t;x) of (1.1) is called time and space periodic solution if it is a solution on
t2R and u(t + T;x) = u(t;x + piei) = u(t;x) for t2R, x2H, and i = 1;2; ;N. It is
called a positive solution if u(t;x) > 0 for all t in the existence interval and x2H.
Proposition 3.8. Consider (1.1) and assume that f0 satis es (H0) and f0(t;x+piei;u) =
f0(t;x;u) (i = 1;2; ;N).
(1) (Uniqueness of periodic positive solutions) If (1.1) has a time and space periodic pos-
itive solution, then it is unique.
(2) (Stability of periodic positive solutions) If (1.1) has a time and space periodic positive
solution u (t;x), then it is globally asymptotically stable with respect to perturbations
in X+p nf0g.
30
(3) (Existence of periodic positive solutions) If (f0( ; ;0)) > 0, then (1.1) has a time and
space periodic positive solution.
Proof. The case that H = H1 and A = A1 follows from the results in [57]. The case that
H = H2 and A = A2 follows from the results in [65]. The case that H = H3 and A = A3
can be proved by the similar arguments as in [65].
Corollary 3.1. (1) If ( ^f0( ;0)) > 0, then (1.1) has a time and space periodic positive
solution.
(2) If ( ^^f0(0)) > 0, then (1.1) has a time and space periodic positive solution.
Proof. (1) follows from Propositions 3.6 and 3.8.
(2) follows from Propositions 3.6, 3.7 and 3.8.
Proposition 3.9. Consider (1.1). Assume that f0 satis es (H0), f0(t;x+piei;u) = f0(t;x;u)
(i = 1;2; ;N), and (f0( ; ;0)) > 0. Then for any given 2SN 1, (1.1) has a spatial
spreading speed c 0( ) in the direction of . Moreover,
c 0( ) = inf >0 ; (f0( ; ;0)) (3.20)
and for any c 0. Then for any given 2 SN 1, (1.1) has a spatial
spreading speed c 0( ) in the direction of and
c 0( ;f0) c 0( ^f0) c 0( ^^f0):
Proof. First, by Propositions 3.6, 3.7, and 3.9, c 0( ;f0, c 0( ; ^f0), and c 0( ; ^^f0) exist. More-
over, by Propositions 3.6 and 3.7 again, and by (3.20),
c 0( ;f0) c 0( ^f0) c 0( ^^f0):
3.5.2 KPP equations in spatially periodic media
In this subsection, we recall some additional spatial spreading dynamics of KPP equa-
tions in spatially periodic media.
Consider (1.1). Throughout this subsection, we assume that f0(t;x;u) f0(x;u).
Proposition 3.10 (Spreading speeds). Consider (1.1). Assume that f0(t;x;u) = f0(x;u)
satis es (H0), f0(x+piei;u) = f0(x;u) (i = 1;2; ;N), and (f0( ;0)) > 0. Then for any
given 2SN 1, (1.1) has a spatial spreading speed c 0( ) in the direction of . Moreover,
c 0( ) is of the following spreading features.
(1) For each u0 2X+ satisfying that u0(x) = 0 for x2H with jx j 1,
lim sup
jx j ct;t!1
u(t;x;u0;f0( ; )) = 0 8c> maxfc 0( );c 0( )g:
(2) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2Hwithjx j r,
lim sup
jx j ct;t!1
ju(t;x;u0;f0( ; )) u 0(x;f0( ; ))j= 0
32
for all 0 sup
2SN 1
c 0( ):
(4) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2H withkxk r,
lim sup
kxk ct;t!1
ju(t;x;u0;f0( ; )) u 0(x)j= 0 80 0:
It then su ces to prove that the limit in(4.2) exists for = 0.
Let 0 1 be such that (u0;u (0; )) = ln 0 and
1
0u
(0;x) u0(x) 0u (0;x) 8x2H:
By Proposition 3.4, there is 1 1 such that
limt!1 (u(t; ;u0);u (t; )) = ln 1:
Moreover, by (4.1),
(u(t; ;u0);u (t; )) (u0;u (0; )) = ln 0
and hence
1
0u
(t;x) u0(t;x;u0) 0u (t;x) 8t> 0; x2H: (4.3)
35
If 1 = 1, then for any > 0, there is > 0 such that for t ,
(u(t; ;u0);u (t; )) ln(1 + ):
This implies that
1
1 + u
(t;x) u(t;x;u0) (1 + )u (t;x) 8t ; x2H:
Hence
ju(t;x;u0) u (t;x)j u (t;x) 8t ; x2H:
It then follows that
limt!1ku(t; ;u0) u (t; )k= 0:
Assume 1> 1. By (4.3), there are > 0, M > 0, and > 0 such that
u(t;x;u0) M; u (t;x) M 8t 0; x2H
and
(u(t; ;u0);u (t; )) 8t 0:
By Proposition 3.4 again, there is > 0 such that for any n 1,
(u(nT; ;u0);u (nT; )) (u((n 1)T; ;u0);u ((n 1)T; ))
and hence
(u(nT; ;u0);u (nT; )) (u0;u (0; )) n 8n 1:
Let n!1, we have
limn!1 (u(nT; ;u0);u (nT; )) = 1:
36
This is a contradiction.
Therefore, we must have 1 = 1 and
limt!1ku(t; ;u0) u (t; )k= 0:
4.2 Existence
In this subsection, we show the existence of time periodic strictly positive solutions of
(1.3), i.e., show Theorem 2.1(3). To this end, we rst prove some lemmas.
Throughout this subsection, we assume the conditions in Theorem 2.1(3). Then by
Proposition 3.8, (1.1) has a unique time and space periodic positive solution u 0(t;x). Let
0 > 0 be such that
0 < 0 < inf
(t;x)2R H
u 0(t;x):
Let 0 : R!R+ be a non-increasing smooth function such that
0 0( ) 0; lim infr! 1 0(r) > 0; 0(r) = 0 8r 1: (4.4)
Lemma 4.1. For give 2SN 1, let u0(x) = 0(x ) and un; (x) = u0(x + n ) (n2N).
There are K 0 and n 0 such that
u(KT; ;un ; ;f) un ; ( ):
Proof. Let > 0 be such that
0 < inf
(t;x)2R H
u 0(t;x) :
37
Fix 0 0
such that u(t;x;un ; ) ~ 0 for t KT and x M.
Proof. By Proposition 3.1 and Lemma 4.1,
u(mKT; ;un ; ;f) un ; ( ) 8m 1: (4.9)
39
Moreover, by the arguments of Lemma 4.1,
u(mKT;x;un ; ;f) 0 8m 1; x n :
It then su ces to prove that for any M 1,
inf
t2[KT;(K+1)T];x2H;x 2[ n ;M]
u(t;x;un ; ;f) > 0: (4.10)
Suppose that (4.10) does not hold. Then there are tn 2 [KT;(K + 1)T] and xn 2H
with xn 2[ n ;M], kxnk!1 such that
u(tn;xn;un ; ;f)!0 as n!1: (4.11)
Without loss of generality, we may assume that tn!t as n!1for some t 2[KT;(K +
1)T]. We then have
ku(tn; ;un ; ;f) u(t ; ;un ; ;f)k!0 as n!1
and hence
u(tn;xn;un ; ;f) u(t ;xn;un ; ;f)!0 as n!1:
Observe that
u(t ;xn;un ; ;f) = u(t ;0;un ; ( +xn);f( ; +xn; )) 8n 1
and
f(t;x+xn;u) f0(t;x+xn;u)!0
40
as n!1 uniformly in (t;x;u) on bounded sets. Observe also that there is n n such
that
un ; ( +xn) un ; ( ) 8n 1:
By Propositions 3.1 and 3.3, we have
u(t ;0;un ; ( +xn);f( ; +xn; )) u(t ;0;un ; ( );f( ; +xn; ))
and
u(t ;0;un ; ( );f( ; +xn; )) u(t ;0;un ; ( );f0( ; +xn; ))!0 (4.12)
as n!1. Without loss of generality, we may also assume that there is ~x2H such that
f0(t;x+xn;u)!f0(t;x+ ~x;u) as n!1
uniformly in (t;x;u) on bounded sets. Then by Proposition 3.3 again,
u(t ;0;un ; ( );f0( ; +xn; ))!u(t ;0;un ; ( );f0( ; + ~x; )) as n!1: (4.13)
By Proposition 3.1,
u(t ;0;un ; ( );f0( ; + ~x; )) > 0: (4.14)
It then follows from (4.12), (4.13), and (4.14) that
lim infn!1 u(tn;xn;un ; ;f) > 0;
which contradicts to (4.11). Therefore, (4.10) holds and the lemma thus follows.
Observe that for any M M0, u(t;x) M is a supersolution of (1.3) on H. Hence
u(T;x;M;f) M
41
and then by Proposition 3.1, u(nT;x;M;f) decreases at n increases. De ne
u+(x) := limn!1u(nT;x;M;f): (4.15)
Then u+(x) is a Lebesgue measurable and upper semi-continuous function. In the following,
we x an M maxfM0; 0g.
Lemma 4.3. There exists > 0 such that u+(x) for x2RN.
Proof. Let 0( ) be as in (4.4) and
u i(x) = 0( x ei); i = 1;2; ;N:
By Proposition 3.1,
u(t; ;M;f) u(t; ;u i;f)
for t 0 and i = 1;2; ;N. By Lemma 4.2, there is and T > 0 such that
u(t;x;u i;f) 8t T; x ei 0; i = 1;2; ;N:
It then follows that
u(t;x;M;f) 8t T; x2H:
This implies that
u(mT;x;M;f) 8m 1; x2H
and then
u+(x) 8x2H:
The lemma thus follows.
Now we prove the existence of time periodic positive solutions
42
Proof of Theorem 2.1(3). We rst claim that
(u(nT; ; =2);u(nT; ;M))!0 (4.16)
as n!1. Assume this is not true. Let
n = (u(nT; ; =2);u(nT; ;M))
and
1 = limn!1 n
(the existence of this limit follows from Proposition 3.4). Then 1> 0,
e 0
1
e 0u(nT; ;M) u(nT; ;
=2) e 0u(nT; ;M) e 0M
for n = 0;1;2; and
(u(nT; ; =2);u(nT; ;M)) 1
for n = 1;2; . By Proposition 3.4, there is > 0 such that
(u(nT; ; =2);u(nT; ;M)) 0 n 8n = 1;2; :
This implies that 1 = 1, a contradiction. Therefore, (4.16) holds.
By (4.16), there is K1 1 such that
=2 u(K1T; ; =2)
and then
=2 u(nK1T; ; =2) u(nK1T; ;M) M 8n = 1;2; :
43
It then follows that
u(nK1T;x;M) u+(x) u(nK1T;x; =2) 8x2H; n = 1;2; :
Therefore
0 u(nK1T;x;M) u+(x) u(nK1T;x;M) u(nK1T;x; =2)
u(nK1T;x;M)(1 1e
n
)
M(1 1e
n
):
This implies that
limn!1u(nK1T;x;M) = u+(x)
uniformly in x2H and u+( )2X++. Moreover, by
u(nK1T; ;M) u(kT; ;M) u((n+ 1)K1; ;M) 8nK1 k (n+ 1)K1;
we have
lim
k!1
u(kT;x;M) = u+(x)
uniformly in x2H and then
u(T; ;u+) = u+( ):
This implies that u (t;x) = u(t;x;u+) is a time periodic strictly positive solutions of (1.3).
4.3 Tail property
In this section, we prove the tail property of time periodic strictly positive solutions of
(1.3). Throughout this subsection, we assume the conditions in Theorem 2.1 (4).
44
Proof of Theorem 2.1 (4). Suppose that u (t;x) is a time periodic strictly positive solution
of (1.3). Observe that u (t;x) = u(t;x;u+), where u+ is as in the proof of Theorem 2.1(3).
We claim
lim
r!1
sup
x2H;kxk r
ju (t;x) u 0(t;x)j= 0: (4.17)
To prove (4.17), we rst show that
limr!1 sup
x2H;kxk r
ju+(x) u+0 (x)j= 0: (4.18)
Recall
u+(x) := limn!1u(nT;x;M;f)
and
u+0 (x) := limn!1u(nT;x;M;f0):
Assume (4.18) is not true. Then there exists 0 > 0 andfxng2R withkxkk!1such
that
ju+(xk) u+0 (xk)j> 0
for k 1.
Since both
u(nT;x;M;f)!u+(x)
and
u(nT;x;M;f0)!u+0 (x)
uniformly on x2H, there is N such that for n N,
ju+(nT;xk;M;f) u+(nT;xk;M;f0)j> 0 8k 1: (4.19)
45
Note that there is ~x0 2H such that
f0(t;x+xk;u)!f0(t;x+ ~x0;u)
as k!1 uniformly in (t;x;u) on bounded sets. Note also that
f(t;x+xk;u) f0(t;x+xk;u)!0
as k!1 uniformly in (t;x;u) on bounded sets. Hence
f(t;x+xk;u)!f0(t;x+ ~x0;u)
as k!1 uniformly in (t;x;u) on bounded sets. Then by Proposition 3.3,
ju(NT;xk;M;f) u(NT;xk;M;f0)j
=ju(NT;0;M;f( ; +xk; )) u(NT;0;M;f0( ; +xk; ))j
ju(NT;0;M;f( ; +xk; )) u(NT;0;M;f0( ; + ~x0; ))j
+ju(NT;0;M;f0( ; + ~x0; )) u(NT;0;M;f0( ; +xk; ))j
!0
as k!1. This contradicts to (4.19). Therefore, (4.18) holds.
Now we prove (4.17). Recall that
u (t;x) = u(t;x;u+;f)
and
u 0(t;x) = u(t;x;u+0 ;f0):
46
Suppose that (4.17) does not hold for some t> 0. Then there are xk 2H with kxkk!1
and 0 > 0 such that
ju(t;xk;u+;f) u(t;xk;u+0 ;f0)j 0 8k 1:
Hence
ju(t;xk;u+;f) u(t;x;u+0 ;f)j=ju(t;0;u+( +xk);f( +xk)) u(t;0;u+0 ( +xk);f0( +xk))j
0
for all k 1. By (H1), (4.18), Proposition 3.3, and the arguments in the proof of (4.18),
lim
k!1
[u(t;0;u+( +xk);f( +xk)) u(t;0;u+0 ( +xk);f0( +xk))] = 0:
This is a contradicts again. Therefore, (4.17) holds.
47
Chapter 5
Spatial Spreading Speeds
In this chapter, we investigate the spatial spreading speeds of (1.3) and prove Theorem
2.2. We rst prove a Lemma.
Throughout this section, we assume the conditions in Theorem 2.2. Let u 0(t;x) be the
unique time and space periodic positive solution of (1.1). Let 0 > 0 be such that
0 < 0 < inf
(t;x)2R H
u 0(t;x):
Lemma 5.1. Let 2SN 1, c> 0 and u0 2X+ be given. If
lim inf
x ct;t!1
u(t;x;u0;f) > 0;
then for any 0 0. Then there are and T > 0 such that
u(t;x;u0;f) 8(t;x)2R+ H; x ct; t T :
Assume that the conclusion of (1) is not true. Then there are 0 < c0 < c, 0 > 0, xn 2H,
and tn2R+ with xn c0tn and tn!1 such that
48
ju(tn;xn;u0;f) u (tn;xn)j 0 8n 1: (5.1)
Note that there are kn 2 Z+ and n 2 [0;T] such that tn = knT + n. Without loss of
generality, we may assume that
n! and xn!x
as n !1 in the case that fkxnkg is bounded (this implies that f(t + tn;x + xn;u) !
f(t+ ;x+x ;u) uniformly in (t;x;u) in bounded sets) and
f(t+tn;x+xn;u) f0(t+ ;x+xn;u)!0
as n!1 uniformly in (t;x;u) on bounded sets in the case that fkxnkg is unbounded.
Let ~u0 2X+,
~u0(x) = 8x2H:
Let
M = sup
x2H
u0(x):
By Theorem 2.1, there is ~T > 0 such that
ju(t;x;M;f) u (t;x)j< 02 8t ~T; x2H; (5.2)
ju( ~T;x; ~u0;f( + ; +x ; )) u ( ~T + ;x+x )j< 02 ; 8x2H; 2R (5.3)
and
ju( ~T;x; ~u0;f0( + ; ; )) u 0( ~T + ;x)j< 02 8x2H; 2R: (5.4)
Without loss of generality, we may assume that tn ~T T for n 1. Let ~u0n2X+
be such that
~u0n(x) = for x c
0 +c
2 (tn
~T);
49
0 ~u0n(x) for c
0 +c
2 (tn
~T) x c(tn ~T);
and
~u0n(x) = 0 for x c(tn ~T):
Then
u(tn ~T; ;u0;f) ~u0n( )
and hence
u(tn;xn;u0;f) = u( ~T;xn;u(tn ~T; ;u0;f);f( +tn ~T; ; ))
= u( ~T;0;u(tn ~T; +xn;u0;f);f( +tn ~T; +xn; ))
u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; )): (5.5)
Observe that
~u0n(x+xn)! ~u0(x)
as n!1 uniformly in x on bounded sets. In the case that
f(tn +t;x+xn;u) f0(t+ ;x+xn;u)!0
as n!1, by Proposition 3.3,
u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; )) u( ~T;0; ~u0;f0( + ~T; +xn; ))!0
as n!1. Then by (5.4) and (5.5),
u(tn;xn;u0;f) >u 0( ;xn) 0=2 for n 1: (5.6)
By Theorem 2.1(4),
50
u 0( ;xn) >u ( ;xn) 0=2 for n 1: (5.7)
By Proposition 3.1 and (5.2),
u(tn;xn;u0;f) u(tn;xn;M;f) u (tn;xn) + 0 8n 1:
(5.6), (5.7), and the continuity of u (t;x) imply,
ju(tn;xn;u0;f) u (tn;xn)j< 0 for n 1:
This contradicts to (5.1).
In the case that xn!x , by Proposition 3.3 again,
u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; ))!u( ~T;0; ~u0;f( + ~T; +x ; ))
as n!1. By (5.3) and (5.5),
u(tn;xn;u0;f) >u ( ;x ) 0=2 for n 1: (5.8)
By the continuity of u ( ; ),
u ( ;x ) >u ( n;xn) 0=2 for n 1: (5.9)
By Proposition 3.1 and (5.2),
u(tn;xn;u0;f) u(tn;xn;M;f) u (tn;xn) + 0 8n 1:
This together with (5.8), and (5.9) implies that
ju(tn;xn;u0;f) u (tn;xn)j< 0 for n 1:
51
This contradicts to (5.1) again.
Hence
lim sup
x c0t;t!1
ju(t;x;u0;f) u (t;x)j= 0
for all 0 0 such that
c 0( ) = ; ( )(f0( ; ;0)) ( )
and
; (f0( ; ;0))
>c
0( ) 8 0 < <
( ):
We now prove Theorem 2.2.
Proof of Theorem 2.2. We rst show that for any c>c 0( ),
lim sup
x ct;t!1
u(t;x;u0;f) = 0: (5.10)
Let 0 < < ( ) be such that
c = ; (f0( ; ;0)) :
By Proposition 3.5, for any > 0, there are > 0 and a( ; )2Xp such that
a(t;x) f0(t;x;0) + ;
c 0( ) < ; (a) < ; (f0( ; ;0)) 0 such that
f(t;x;u) f(t;x;0) a(t;x) 8t2R; x ~M; u 0: (5.12)
Let M ku0k be such that
f(t;x;uM(t;x)) a(t;x) 8t 0; x ~M: (5.13)
It then follows from (5.11), (5.12), and (5.13) that uM(t;x) is a super-solution of (1.3). By
Proposition 3.1,
u(t;x;u0;f) uM(t;x) = Me (x
; (a)
t) (t;x) 8t 0; x2H:
This implies that (5.10) holds.
Next, we prove that for any c 0 such that
c< inf >0 ; (f0( ; ;0) ) 0 such that
f(t;x;u) f0(t;x;u) 8t2R; x M; 0 u 1: (5.15)
By Lemma 4.2, there are ~ > 0 and ~T > 0 such that
u(t;x;u0;f) ~ 8t ~T; x M: (5.16)
Consider equation
ut = (Au)(t;x) + [f0(t;x;u) K u]u(x); x2H (5.17)
By Proposition 3.8, (5.17) has a unique time and space periodic solution u 0;K (t;x;). Let
K 1 be such that
u0;K ( ~T;x) ~ :
Let ~u0 2X+( ) be such that
~u0(x) minfu 0;K ( ~T;x);u( ~T;x;u0;f)g:
By Proposition 3.1,
u(t+ ~T;x;u0;f) u(t;x; ~u0; ~f0( + ~T; ; )) 8t 0; x M;
where ~f0(t;x;u) = f0(t;x;u) K u. By Proposition 3.9, for any c0 ; (f0 ) ,
lim sup
x c0t;t!1
ju(t;x; ~u0; ~f0( +~; ; )) u 0;K (t+ ~T;x)j= 0: (5.18)
54
By (5.16) and (5.18),
lim inf
x c0t;t!1
u(t;x;u0;f) > 0:
This together with Lemma 5.1 implies that (5.14) holds.
55
Chapter 6
Another Method to Show the Time Independent Case
In this chapter, we consider time independent monostable equations and provide some
other method to prove Theorems 2.1 and 2.2. We also show Theorem 2.3.
6.1 Another method to prove Theorem 2.1
In this section, we investigate the existence of positive stationary solutions of (1.3) in
the special case that f(t;x;u) f(x;u) and f0(x;u) f0(u) and give another proof of
Theorem 2.1.
Throughout this section, we assume f(t;x;u) f(x;u), f0(x;u) f0(u) and (H0),
(H1)0 and (H2). We rst prove some lemmas.
For convenience, we denote f(x;u) and f0(u) by fi(x;u) and f0i (u), respectively, in the
case H=Hi and A=Ai for i = 1;2;3. We may write u(t;x;u0;fi) as ui(t;x;u0).
Lemma 6.1. For any 1 i 3 and > 0, there are p = (p1;p2; ;pN) 2 NN and
hi2Xi;p\CN(Hi;R) such that
fi(x;0) hi(x) for x2Hi;
^^h
i f0i (0) (hence (hi( )) f0i (0) );
and for the cases that i = 1 and 2, the partial derivatives of hi(x) up to order N 1 are zero
at some x0 2Hi with hi(x0) = maxx2Hi(x), where ^^hi is the average of hi( ) (see (3.16) for
the de nition).
56
Proof. Fix 1 i 3. By (H1)0, there are 0 < 0 1 and L0 > 0 such that
fi(x;0) f0i (0) 0 for x2Hi; kxk L0:
Let
M0 = inf
x2Hi;1 i 3
fi(x;0):
Let h0 : R![0;1] be a smooth function such that
h0(s) =
8
>>>
>>><
>>>
>>>:
1 for jsj 1
0 for jsj 2:
For any p = (p1;p2; ;pN)2NN with pj > 4L0, let hi2Xi;p\CN(Hi;R) (i = 1;2;3) be
such that
hi(x) = f0i (0) 0 h0 kxk
2
L20
(f0
i (0) 0 M0)
for x2
[ p12 ;p12 ] [ p22 ;p22 ] [ pN2 ;pN2 ]
\Hi:
Then
fi(x;0) hi(x) 8x2Hi; 1 i 3:
It is clear that for i = 1 or 2, the partial derivatives of hi(x) up to order N 1 are zero at
some x0 2Hi with hi(x0) = maxx2Hihi(x)(= f0i (0) 0). For given > 0, choosing pj 1,
we have
^^h
i >f0i (0) :
By Proposition 3.7, (hi( )) (^^hi) = ^^hi and hence
(hi( )) f0i (0) :
57
The lemma is thus proved.
Lemma 6.2. Suppose that ~u 2 : RN ![ 0;M0] is Lebesgue measurable, where 0 and M0 are
two positive constants. If
Z
RN
(y x)~u 2(y)dy ~u 2(x) + ~u 2(x) ~f2(x; ~u 2(x)) = 0 8x2RN;
where ~f2(x;u) = f2(x;u) or f02 (u) for all x2RN and u2R, then ~u 2( )2X++2 .
Proof. We prove the case that ~f2(x;u) = f2(x;u). The case that ~f2(x;u) = f02 (u) can be
proved similarly.
Let
h (x) =
Z
RN
(y x)~u 2(y)dy for x2RN:
Then h ( ) is C1 and has bounded rst order partial derivatives. Let
F(x; ) = h (x) + f2(x; ) 8x2RN; 2R:
Then F : RN R!R is C1 and F(x; ~u 2(x)) = 0 for each x2RN. If > 0 is such that
F(x; ) = 0, then
1 +f2(x; ) = h
(x)
< 0
and hence
@ F(x; ) = 1 +f2(x; ) + @uf2(x; ) < 0:
By Implicit Function Theorem, ~u 2(x) is C1 in x. Moreover,
@~u 2(x)
@xj =
@h (x)
@xj
1 +f(x; ~u 2(x)) +@uf2(x; ~u 2(x))~u 2(x) 8x2R
N; 1 j N:
Therefore, ~u 2 has bounded rst order partial derivatives. It then follows that ~u 2(x) is uni-
formly continuous in x2RN and then ~u 2 2X++2 .
58
Lemma 6.3. Suppose that u i( )2X++i and u = u i( ) is a stationary solution of (1.3) with
H=Hi, A=Ai, and f(t;x;u) = fi(x;u). Then
u i(x)!u0i as kxk!1;
where u0i is a positive constant such that f0i (u0i) = 0.
Proof. We rst prove that
u 1(x)!u01 as kxk!1:
Assume that u 1(x)6!u01 as kxk!1. Then there are 0 > 0 and xn2RN such that
kxnk!1
and
ju 1(xn) u01j 0 for n = 1;2; :
By the uniform continuity of u 1(x) in x2RN, without loss of generality, we may assume
that there is a continuous function ~u 1 : RN ![ 0;M0] for some 0;M0 > 0 such that
u1(x+xn)! ~u 1(x)
as n!1 uniformly in x on bounded sets. Moreover, by a priori estimates for parabolic
equations, ~u 1 is C2+ for some > 0 and we may also assume that
u1(x+xn)! ~u 1(x)
as n!1 uniformly in x on bounded sets. This together with f1(x + xn;u) !f01 (u) as
n!1 uniformly in x on bounded sets and in u2R implies that
~u 1 + ~u 1f01 (~u 1) = 0; x2RN:
59
By Proposition 3.8, we must have
~u 1(x) u 1(x;f01 ( )) u01
and hence
u 1(xn)!u01 as n!1:
This is a contradiction. Therefore
u 1(x)!u01 as kxk!1:
Next, we prove that
u 2(x)!u02 as kxk!1:
Similarly, assume that u 2(x) 6!u02 as kxk!1. Then there are 0 > 0 and xn 2RN such
that kxnk!1 and
ju 2(xn) u02j 0 for n = 1;2; :
By the uniform continuity of u 2(x) in x2RN, without loss of generality, we may assume
that there is a continuous function ~u 2 : RN ![ 0;M0] for some 0;M0 > 0 such that
u 2(x+xn)! ~u 2(x)
as n ! 1 uniformly in x on bounded sets. By the Lebesgue Dominated Convergence
Theorem, we have
Z
RN
(y x)~u 2(y)dy ~u 2(x) + ~u 2(x)f02 (~u 2(x)) = 0 8x2RN:
By Lemma 6.2, ~u 2 2X++2 . By Proposition 3.8 again, we have ~u 2(x) u02 and then u 2(xn)!
u02 as n!1. This is a contradiction. Therefore u 2(x)!u02 as kxk!1.
60
Finally, it can be proved by the similar arguments as in the case i = 2 that
u 3(j)!u03 as kjk!1:
Lemma 6.4. There is u i 2 X++i such that for any > 0 su ciently small, u(t;x; u i )
is increasing in t > 0 and u ; ; 2X++i , where u ; ; (x) = limt!1u(t;x; u i ), and hence
u = u ; ; ( ) is a stationary solution of (1.3) in X++i in the case H = H1, A = Ai, and
f(t;x;u) = fi(x;u) (i = 1;2;3).
Proof. Fix 1 i 3. Let M > 0 be such that fi(x;M ) < 0. Let > 0 be such that
f0i (0) > 0:
By Lemma 6.1, there are p2NN and hi( )2Xi;p\CN(Hi;R) such that
fi(x;0) hi(x); and ^hi f0i (0) (> 0):
Moreover, for i = 1 or 2, the partial derivatives of hi(x) up to order N 1 are zero at some
x0 2Hi with hi(x0) = maxx2Hihi(x). Let u i be the positive principal eigenfunction of
Ai +hi( )I withku ik= 1 (the existence of u i is well known in the case that i = 1 or 3 and
follows from Proposition 3.5 in the case that i = 2). It is not di cult to verify that u = u i
is a sub-solution of (1.3) for any > 0 su ciently small. It then follows that for any > 0
su ciently small,
u i ( ) u(t1; ; u i ) ui(t2; ; u i ) 80 0 such that
limt!1u(t;x; u i ) = u ; ; i (x) 8x2Hi:
Moreover, by regularity and a priori estimates for parabolic equations, u ; ; 1 2X++1 . It is
clear that u ; ; 3 2X++3 . By Lemma 6.2, u ; ; 2 2X++2 . Therefore for 1 i 3, u ; ; i 2X++i
and u = u ; ; i ( ) is a stationary solution of (1.3) in X++i (i = 1;2;3).
Lemma 6.5. Let M 1 be such that fi(x;M) < 0 for x 2 Hi (i = 1;2;3). Then
limt!1u(t;x;u0) exists for every x2Hi, where u0(x) M. Moreover, u+; ;Mi ( ) 2X++i ,
where u+; ;Mi (x) := limt!1ui(t;x;u0), and hence u = u+; ;Mi ( ) is a stationary solution of
(1.3) in X++i in the case H=Hi and A=Ai (i = 1;2;3).
Proof. Fix 1 i 3. For any M > 1 with fi(x;M) < 0 for all x 2Hi, u = M is a
super-solution of (1.3). Hence
u(t2; ;M) u(t1; ;M) M 80 t1 1 such that
i(u1; i ;u2; i ) = ln > 0:
Note that
1
u
1;
i u
2;
i
u1;
i :
By Lemma 6.3,
lim
kxk!1
u1; i (x) = u0i
and
lim
kxk!1
u2; i (x) = u0i:
This implies that there is > 0 such that
1
u
1;
i (x) u
2;
i (x) (
)u1;
i (x) for kxk 1:
By Proposition 3.1 and the arguments in Proposition 3.4,
1
u
1;
i (x) __ 0 su ciently small
and M > 0 su ciently large such that u i u0 M and u = u i is a sub-solution of (1.3)
(u i is as in Lemma 6.4) and u = M is a super-solution of (1.3). Then
u i ui(t; ; u i ) ui(t; ;u0) ui(t; ;M) M 8t 0:
By (1), Lemmas 6.4 and 6.5, and Dini?s Theorem,
ui(t;x; u i ) ____ 0; x2Hi
and
limt!1ui(t;x; u i ) = limt!1ui(t;x;M) = u i(x)
uniformly in x on bounded sets. It then follows that
limt!1ui(t;x;u0) = u i(x)
uniformly in x on bounded sets.
We claim that kui(t; ;u0) u i( )k!0 as t!1. Assume the claim is not true. Then
there are 0 > 0, tn!1, and xn with kxnk!1 such that
jui(tn;xn;u0) u i(xn)j 0 8n2N:
Then by Lemma 6.3,
jui(tn;xn;u0) u0ij 02 8n 1:
Let ~ > 0 and ~M > 0 be such that
~ ui(t; ;u0) ~M 8t 0:
64
For any > 0, let T > 0 be such that
jui(T; ; ~ ;f0i ( )) u0ij< ; jui(T; ; ~M;f0i ( )) u0ij< : (6.1)
Observe that
~ ui(tn T;xn +x;u0) ~M
and
ui(tn;xn + ;u0) = ui(T;xn + ;ui(tn T; ;u0)) = ui(T; ;ui(tn T; +xn;u0);fi( +xn; ))
for n 1. Then
ui(T; ; ~ ;fi( +xn)) ui(tn;xn + ;u0) ui(T; ; ~M;fi( +xn; )): (6.2)
Observe also that
fi(x+xn;u)!f0i (u)
as n!1 uniformly in (x;u) on bounded sets. Then by Proposition 3.3,
ui(T;x; ~ ;fi( +xn; ))!ui(T;x; ~ ;f0i ( ))
and
ui(T;x; ~M;fi( +xn; ))!ui(T;x; ~M;f0i ( ))
as n!1 uniformly in x on bounded sets. This together with (6.1) implies that
jui(T;0; ~ ;fi( +xn; )) u0ij< 2 ; jui(T;0; ~M;fi( +xn; )) u0ij< 2 for n 1
65
and then by (6.2),
jui(tn;xn;u0) u0ij< 2 for n 1:
Hence
limn!1ui(tn;xn;u0) = u0i;
which is a contradiction. Therefore
kui(t; ;u0) u i( )k!0
as t!1.
Finally, note that (4) follows from Lemma 6.3.
6.2 Spatial Spreading Speeds and Proofs of Theorems 2.2 and 2.3
In this section, we explore the spreading speeds of (1.3) in the special case thatf(t;x;u) =
f(x;u) and f0(x;u) = f0(u). We provide another proof of Theorem 2.2 and give a proof of
Theorem 2.3.
Throughout this section, we assume f(t;x;u) f(x;u) and f0(x;u) f0(u), and
assume (H0), (H1)0 and (H2).
For convenience again, we denote f(x;u) and f0(u) by fi(x;u) and f0i (u), respectively,
in the case H=Hi and A=Ai for i = 1;2;3. We may write u(t;x;u0;fi) as ui(t;x;u0) or
ui(t;x;u0;fi), write c 0( ;f0i ) as c0i( ), and ; (f0i (0)) as i( ; ;f0i (0)).
We rst prove two lemmas.
Lemma 6.6. Let 2SN 1, c> 0, 1 i 3, and u0 2X+i be given.
(1) If lim infx ct;t!1ui(t;x;u0) > 0, then for any 0 0, then for any 0 0, then for any 0 0. Then there are and T > 0 such
that
ui(t;x;u0) 8(t;x)2R+ Hi; x ct; t T:
Assume that the conclusion of (1) is not true. Then there are 0 < c0 < c, 0 > 0, xn2Hi,
and tn2R+ with xn c0tn and tn!1 such that
jui(tn;xn;u0) u i(xn)j 0 8n 1: (6.3)
Without loss of generality, we may assume that
xn!x
as n!1in the case thatfkxnkgis bounded (this implies that fi(x+xn;u)!fi(x+x ;u)
uniformly in (x;u) in bounded sets) and
fi(x+xn;u)!f0i (u)
as n!1 uniformly in (x;u) on bounded sets in the case that fkxnkg is unbounded.
Let ~u0 2X+i ,
~u0(x) = 8x2Hi:
67
By Theorem 2.1, there is ~T > 0 such that
ui( ~T;x;u0) u i(x) < 0 8x2Hi; (6.4)
jui( ~T;x; ~u0;fi( +x ; )) u i(x+x )j< 02 ; (6.5)
and
jui( ~T;x; ~u0;f0i ) u0ij< 02 : (6.6)
Without loss of generality, we may assume that tn ~T T for n 1. Let ~u0n2X+i be
such that 8
>>>
>>>
>>>>
>>>
><
>>>
>>>>
>>>
>>>>
:
~u0n(x) = for x c0+c2 (tn ~T)
0 ~u0n(x) for c0+c2 (tn ~T) x c(tn ~T)
~u0n(x) = 0 for x c(tn ~T):
Then
ui(tn ~T; ;u0) ~u0n( )
and hence
ui(tn;xn;u0) = ui( ~T;xn;ui(tn ~T; ;u0))
= ui( ~T;0;ui(tn ~T; +xn;u0);fi( +xn; ))
ui( ~T;0; ~u0n( +xn);fi( +xn; )): (6.7)
Observe that
~u0n(x+xn)! ~u0
68
as n!1 uniformly in x on bounded sets. In the case that
fi(x+xn;u)!f0i (u);
by Proposition 3.3,
ui( ~T;0; ~u0n( +xn);fi( +xn; ))!ui( ~T;0; ~u0;f0i ( ))
as n!1. By (6.6) and (6.7),
ui(tn;xn;u0) >u0i 0=2 for n 1: (6.8)
By Lemma 6.3,
u0i >u i(xn) 0=2 for n 1: (6.9)
By (6.4), (6.8), and (6.9),
jui(tn;xn;u0) u i(xn)j< 0 for n 1:
This contradicts to (6.3).
In the case that
xn!x ;
by Proposition 3.3 again,
ui( ~T;0; ~u0n( +xn);fi( +xn; ))!ui( ~T;0; ~u0;fi( +x ; ))
as n!1. By (6.5) and (6.7),
ui(tn;xn;u0) >u i(x ) 0=2 for n 1: (6.10)
69
By the continuity of u i( ),
u i(x ) >u i(xn) 0=2 for n 1: (6.11)
By (6.4), (6.10), and (6.11),
jui(tn;xn;u0) u i(xn)j< 0 for n 1:
This contradicts to (6.3) again.
Hence
lim
x c0t;t!1
jui(t;x;u0) u i(x)j= 0
for all 0 0 be such that fi(x;u) < 0 for x2Hi, u2 [0;M], and i = 1;2;3.
Then for any > 0, there are p 2 NN and gi : Hi R ! R such that for any u 2 R,
gi( ;u)2Xi;p, gi( ; ) satis es (H0) and (H2), and
fi(x;u) gi(x;u) 8x2Hi; u2[0;M];
^gi(0) f0i (0) ;
where ^^gi( ) is as in (2.20) (i = 1;2;3).
Proof. By Lemma 6.1, for any > 0, there are p2NN and hi( ) 2Xi;p\CN(Hi;R) such
that
fi(x;0) hi(x) 8x2Hi
70
and
^^h
i f0i (0)
for i = 1;2;3. Fix 1 i 3 and choose Mi > 0 such that
fi(x;u) gi(x;u) := hi(x) Miu for x2Hi; 0 u M:
It is not di cult to see that gi( ; ) (1 i 3) satisfy the lemma.
Recall that for given 2SN 1,
c01( ) = inf >0 f
0
1 (0) +
2
= 2
q
f01 (0);
c02( ) = inf >0
R
RN e
z (z)dz 1 +f0
2 (0)
;
and
c03( ) = inf >0
P
k2Kak(e
k 1) +f0
3 (0)
:
Observe that i( ; ;f0i (0)) (i = 1;2;3) exist,
1( ; ;f01 (0)) = f01 (0) + 2;
2( ; ;f02 (0)) =
Z
RN
e z (z)dz 1 +f02 (0);
and
3( ; ;f03 (0)) =
X
k2K
ak(e k 1) +f03 (0):
If no confusion occurs, we may denote i( ; ;f0i (0)) by i( ; ) (i = 1;2;3). Observe also
that
v1(t;x) = e (x 1( ; ) t);
v2(t;x) = e (x 2( ; ) t);
71
and
v3(t;j) = e (j 3( ; ) t)
are solutions of
vt(t;x) = v(t;x) +f01 (0)v(t;x); x2RN; (6.12)
vt(t;x) =
Z
RN
(y x)v(t;y)dy v(t;x) +f02 (0)v(t;x); x2RN; (6.13)
and
vt(t;j) =
X
k2K
ak(v(t;j +k) v(t;j)) +f03 (0)v(t;j); j2ZN; (6.14)
respectively.
We now give another proof of Theorem 2.2.
Proof of Theorem 2.2. Fix 2SN 1 and 1 i 3. We rst prove that for any c0 > c0i( )
and u0 2X+i ( ),
lim sup
x c0t;t!1
ui(t;x;u0) = 0: (6.15)
To this end, take a c such that c0 >c>c i( ). Note that there is i > 0 such that
c0i( ) = i( ;
i)
i
and there is 2(0; i) such that
c = i( ; ) :
Take d>M > 0 such that
u0(x) M and u0(x) de x 8x2Hi;
fi(x;M) < 0 8x2Hi; (6.16)
72
and
fi(x;u) = f0i (u) for x 1 ln Md (> 0): (6.17)
Observe that by (6.16) and (H2), for (t;x)2(0;1) Hi with de (x ct) M, i.e., x
1 ln Md +ct,
fi(x;de (x ct)) < 0 0 x2Hi: (6.18)
This implies that (6.15) holds.
Next, we prove that for any c0 0 be such that
u0(x) M and fi(x;M) < 0 for all x2Hi:
73
Then u M is a super-solution of (1.3) and
ui(t;x;u0) M 8t 0; x2Hi:
For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small,
c i( ;gi( ; )) c i( ;^^gi( )) >c:
By Propositions 3.1 and 3.10,
lim inf
x ct;t!1
ui(t;x;u0) lim inf
x ct;t!1
ui(t;x;u0;gi) > 0:
(6.19) then follows from Lemma 6.6.
By (6.15) and (6.19), c i( ) exists and c i( ) = c0i( ) for i = 1;2;3. Moreover, (2.26)
holds
Finally, prove Theorem 2.3.
Proof of Theorem 2.3. (1) Fix 2SN 1 and 1 i 3. Let u0 2X+i satisfy that u0(x) = 0
for x2Hi with jx j 1. Then there are u+0 2X+i ( ) and u 0 2X+i ( ) such that
u0(x) u 0 (x) 8x2Hi:
By Proposition 3.1 and Theorem 2.2,
lim sup
x c0t;t!1
ui(t;x;u0) lim sup
x c0t;t!1
ui(t;x;u+i ) = 0 8c0 >c i( )
and
lim sup
x ( ) c0t;t!1
ui(t;x;u0) lim sup
x ( ) c0t;t!1
ui(t;x;u i ) = 0 8c0 >c i( )
74
It then follows that
lim sup
jx j c0t;t!1
ui(t;x;u0) = 0 8c0 > maxfc i( );c i( )g:
(2) Fix 2SN 1 and 1 i 3. For given 0 < c0 < minfc i( );c i( )g, take a c > 0
such that
c0 0 be such that
u0(x) M and fi(x;M) < 0 for all x2Hi:
Then u M is a super-solution of (1.3) and
ui(t;x;u0) M 8t 0; x2Hi:
For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small,
c i( ;gi( ; )) c i( ;^gi( )) >c:
By Propositions 3.1 and 3.10,
lim inf
jx j ct;t!1
ui(t;x;u0) lim inf
jx j ct;t!1
ui(t;x;u0;gi) > 0:
It then follows from Lemma 6.6 that
lim sup
jx j c0t;t!1
jui(t;x;u0) u i(x)j= 0:
75
(3) Fix 2SN 1 and 1 i 3. Let
c> sup
2SN 1
c i( ):
Let u0 2X+i be such that
u0(x) = 0 for kxk 1:
Note that for every given 2SN 1, there is ~u0( ; )2X+i ( ) such that
u0( ) ~u0( ; ):
By Proposition 3.1,
0 ui(t;x;u0) ui(t;x; ~u0( ; ))
for t> 0 and x2Hi. It then follows from Theorem 2.2 that
0 lim sup
x ct;t!1
ui(t;x;u0) lim sup
x ct;t!1
ui(t;x; ~u0( ; )) = 0:
Take any c0 >c. Consider all x2Hi withkxk= c0. By the compactness of @B(0;c0) =
fx2Hijkxk = c0g, there are 1; 2; ; L2SN 1 such that for every x2@B(0;c0), there
is l (1 l L) such that
x l c:
Hence for every x2Hi with kxk c0t, there is 1 l L such that
x l = kxkc0
c0
kxkx
l kxkc0 c ct:
By the above arguments,
0 lim sup
x l ct;t!1
ui(t;x;u0) lim sup
x l ct;t!1
ui(t;x; ~u0( ; l)) = 0
76
for l = 1;2; L. This implies that
lim sup
kxk c0t;t!1
ui(t;x;u0) = 0:
Since c0 >c and c> sup 2SN 1 c i( ) are arbitrary, we have that for c> sup 2SN 1 c i( ),
lim sup
kxk ct;t!1
ui(t;x;u0) = 0:
(4) It can be proved by similar arguments as in (2). To be more precise, for given
0 0 such that
c0 0 be such that
u0(x) M and fi(x;M) < 0 for all x2Hi:
Then u M is a super-solution of (1.3) and
ui(t;x;u0) M 8t 0; x2Hi:
For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small,
c i( ;gi( ; )) c i( ;^gi( )) >c:
By Propositions 3.1 and 3.10,
lim inf
kxk ct;t!1
ui(t;x;u0) lim inf
kxk ct;t!1
ui(t;x;u0;gi) > 0:
77
It then follows from Lemma 6.6 that
lim sup
kxk c0t;t!1
jui(t;x;u0) u i(x)j= 0:
78
Chapter 7
E ects of Temporal and Spatial Variations
In this chapter, we prove Theorem 2.4 on the e ects of temporal and spatial variations
on the spatial spreading dynamics of monostable equations.
Proof of Theorem 2.4. (1) Suppose that ( ^f0( ;0)) > 0. By Proposition 3.6,
(f0( ; ;0)) ( ^f0( ;0)) > 0:
Then by Theorem 2.1, (1.3) has a time periodic strictly positive solution u (t;x). By The-
orem 2.2, (1.3) has a spatial spreading speed c ( ) in the direction of for each 2SN 1,
and
c ( ) = c 0( ):
Recall that
c 0( ) = inf >0 ; (f0( ; ;0)) :
By Proposition 3.6 again,
; (f0( ; ;0)) ; ( ^f0( ;0)):
It then follows that
c ( ) = c 0( ) inf >0 ; (
^f0( ;0))
= ^c
0( ):
(2) Suppose that ( ^^f0(0)) > 0. By Proposition 3.7, we have
( ^f0( ;0)) ( ^^f0(0)) > 0:
79
It then follows from (1) that (1.3) has a time periodic strictly positive solution u (t;x) and
has a spatial spreading speed c ( ) in the direction of for each 2SN 1.
Recall again that
c 0( ) = inf
>0
; (f0( ; ;0))
:
By Proposition 3.7 again,
; ( ^f0( ;0)) ; ( ^^f0(0)):
This together with (1) implies that
c ( ) = c 0( ) ^c 0( ) ^^c 0( )
for each 2SN 1.
80
Chapter 8
Concluding Remarks and Future Plan
8.1 Concluding Remarks
In this dissertation, we studied the semilinear dispersal evolution equations of the form
ut(t;x) = (Au)(t;x) +u(t;x)f(t;x;u(t;x)); x2H;
where H = RN or ZN, A is a random dispersal operator or nonlocal dispersal operator in
the case H= RN and is a discrete dispersal operator in the case H= ZN, and f is periodic
in t, asymptotically periodic in x (i.e. f(t;x;u) f0(t;x;u) converges to 0 as kxk!1 for
some time and space periodic function f0(t;x;u)), and is of KPP type in u. It is proved
that Liouville type property for such equations holds, that is, time periodic strictly positive
solutions are unique. It is also proved that if u 0 is a linearly unstable solution to the
time and space periodic limit equation of such an equation, then it has a unique stable
time periodic strictly positive solution and has a spatial spreading speed in every direction.
Moreover, we developed multiple methods to achieve the two results, and considered the
e ects of temporal and spatial variations of the media on the uniform persistence and spatial
spreading speeds of monostable equations.
It should be pointed out that the Liouville type property for (1.3) in the case that
H = RN, A = u, and f(t;x;u) = f(x;u) has been proved in [8]. However, many tech-
niques developed in [8] are di cult to apply to (1.3). Several important new techniques
are developed in the current dissertation. Both the results and techniques established in
the current paper can be extended to more general cases (say, cases that A is some linear
81
combination of random and nonlocal dispersal operators and/or f(t;x;u) is almost periodic
in t and asymptotically periodic in x).
It should also be pointed out that, if u = 0 is a linearly unstable solution of (1.1) with
respect to periodic perturbations, then (1.1) has traveling wave solutions connecting 0 and
u 0( ; ) and propagating in the direction of with speed c>c 0( ) for any 2SN 1. But (1.3)
may have no traveling wave solutions connecting 0 and u ( ; ) (see [63]) (hence, localized
spatial inhomogeneity may prevent the existence of traveling wave solutions).
8.2 Future plans
8.2.1 Single-species population model
For the single-species model, we established the existence of strictly positive solution u
and spreading speeds. Next topic I am interested in is to generalize traveling wave solution,
which is a global-in-time solution, connecting the positive solution u and trivial solution
u = 0, in the locally inhomogeneous media. Due to the spatial heterogeneity, the global-in-
time wave-like solutions lose some classical features, compared to homogeneous equations,
such as, constant in shape (up to a spatial shift) and the unchanged speed for each pro le.
Recently, James Nolen together his collaborates considered (1.3) in temporal independent
setting, and showed that a su ciently strong localized inhomogeneity may prevent existence
of global-in-time wave-like solutions. They created a time-global bump-like solution, which
blocks any wave-like solution. I am curious to know whether such interesting scenario happen
in more general heterogeneous media (say, a spatially locally perturbed spatial periodic
media). How about the existence of wave-like solutions in nonlocal and discrete dispersal
framework?
As a short term plan, the following problem is listed in my agenda.
Explore the existence, uniqueness and stability of of global-in-time wave-like solutions
in spatially locally perturbed media.
82
8.2.2 Multi-species population model
Comparing with single species model, much less is known about the multi-species system,
especially in the heterogeneous (say, spatially locally inhomogeneous) environments.
For instance, for two species competitive system, main results established in the litera-
ture are related to spatial-temporal independent nonlinearities,
8
>>>>
>><
>>>>
>>:
@u
@t = u+uf(u;v); x2R
N;
@v
@t = v +vg(u;v); x2R
N;
where f(u;v) < 0 and g(u;v) < 0 for u;v 0 with u2 + v2 1, fu(u;v) < 0, fv(u;v) < 0,
gu(u;v) < 0, and gv(u;v) < 0 for u;v 0 (see [28], [36], [38], [47], [48], [53], etc.)
Here are three problems being considered.
Explore the existence of traveling wave solutions to the systems in periodic media with
nonlocal dispersal.
Extend the concept of spreading speeds for multi-species system in homogeneous and
periodic media to general inhomogeneous media.
Investigate the existence of spreading speeds to the systems in the locally inhomoge-
neous media.
To attack the above problems, we will rst extend the existing results on uniform per-
sistence, coexistence, and extinction of two species competition systems in homogeneous and
periodic media (see [15], [33], [34], etc.) to locally spatially inhomogeneous media.
Regarding the long term research plan, I have following problems in mind.
Study the monostable equations with more general nonlinearity, that is almost periodic
or random stationary ergodic in time.
83
Investigate the dynamics of other types of equations, such as ignition type equations
and bistable type equations which arise in Ising and combustion models and phase
transition models.
84
Bibliography
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bustion, and nerve pulse propagation, in \Partail Di erential Equations and Related
Topics" (J. Goldstein, Ed.), Lecture Notes in Math., Vol. 466, Springer-Verlag, New
York, 1975, pp. 5-49.
[2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear di usions arising in
population genetics, Adv. Math., 30 (1978), pp. 33-76.
[3] P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to
a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332
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