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 Bibliography [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear di usions arising in population genetics, Adv. Math., 30 (1978), pp. 33-76. [2] 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 (2007), pp. 428-440. [3] H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc. 7 (2005), pp. 172-213. [4] H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, II - General domains, J. Amer. Math. Soc. 23 (2010), no. 1, pp. 1-34 [5] H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environ- ment model: II - Biological invasions and pulsating traveling fronts, J. Math. Pures Appl. 84 (2005), pp. 1101-1146. [6] Reinhard B urger, Perturbations of positive semigroups and applications to population genetics, Math. Z. 197 (1988), pp. 259-272. [7] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal di usion equations, J. Math. Pures Appl., 86 (2006), pp. 271-291. [8] C. Cortazar, M. Elgueta, and J. D. Rossi, Nonlocal di usion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. of Math., 170 (2009), pp. 53-60. [9] C. Cortazar, M. Elgueta, Manuel, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal di usion problems, Arch. Ration. Mech. Anal. 187 (2008), pp. 137-156. [10] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction di usion equation, Annali di Matematica 185(3) (2006), pp. 461-485 [11] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Di erential Equations 249 (2010), pp. 2921-2953. [12] J. Coville, J. D avila, and S. Mart nez, Existence and uniqueness of solutions to a non- local equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), pp. 1693- 1709. 83 [13] J. Coville, Jrme, J. Dvila, and S. Martnez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincar Anal. Non Linaire 30 (2013), pp. 179-223. [14] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction- di usion equations, Nonlinear Analysis 60 (2005), pp. 797 - 819. [15] L. C. Evans, Partial Di erential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [16] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003. [17] R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7(1937), pp. 335-369. [18] J. Garc a-Mel an and J. D. Rossi, On the principal eigenvalue of some nonlocal di usion problems, J. Di erential Equations, 246 (2009), pp. 21-38. [19] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local disper- sal, Di erential Integral Equations, 18 (2005), pp. 1299-1320. [20] F. Hamel, Qualitative properties of monostable pulsating fronts : exponential decay and monotonicity, J. Math. Pures Appl. 89 (2008), pp. 355-399. [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981. [22] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes Math. 247, Longman, New York, 1991. [23] G. Hetzer, W. Shen, and A. Zhang, E ects of spatial variations and dispersal strate- gies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain Journal of Mathematics. 43 (2013), pp. 489-513. [24] J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dynam. Syst., 8 (2009), pp. 790-821. [25] W. Hudson and B. Zinner, Existence of traveling waves for reaction di usion equations of Fisher type in periodic media, Boundary value problems for functional-di erential equations, 187{199, World Sci. Publ., River Edge, NJ, 1995. [26] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math 17 (2006), pp. 221-232. [27] V. Hutson, S. Martinez, K. Mischaikow, and G.T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), pp. 483-517. [28] V. Hutson, W. Shen and G.T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain Journal of Mathematics 38 (2008), pp. 1147-1175. 84 [29] Y. Kametaka, On the nonlinear di usion equation of Kolmogorov-Petrovskii- Piskunov type, Osaka J. Math., 13 (1976), pp. 11-66. [30] C.-Y. Kao, Y. Lou, and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), no. 2, pp. 551-596 [31] A. Kolmogorov, I. Petrowsky, and N.Piscunov, A study of the equation of di usion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ., 1 (1937), pp. 1-26. [32] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semi ows with applications, Comm. Pure Appl. Math., 60 (2007), no. 1, pp. 1-40. [33] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), pp. 857-903. [34] W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), pp. 2302-2313. [35] R. Lui, Biological growth and spread modeled by systems of recursions, Math. Bio- sciences, 93 (1989), pp. 269-312. [36] G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-di usion equations, J. Math. Anal. Appl., 385 (2012), pp. 1094-1106. [37] G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction- di usion equations, J. Math. Pures Appl. (9) 98 (2012), pp. 633-653. [38] J. Nolen, M. Rudd, and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), pp. 1-24. [39] J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear ows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), pp. 1217-1234. [40] S. Pan, W.-T. Li, and G. Lin, Existence and stability of traveling wavefronts in a nonlocal di usion equation with delaystar, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), pp. 3150-3158. [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di erential Equa- tions, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983. [42] Nar Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigen- values of time oeriodic nonlocal dispersal operators and applications, Journal of Dy- namics and Di erential Equations, 24 (2012), pp. 927-954. [43] Nar Rawal, W. Shen, and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, submitted. 85 [44] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), pp. 312-355. [45] H. Schaefer, Some spectral properties of positive linear operators, Paci c J. Math. 10 (1960), pp. 1009-1019. [46] W. Shen, Variational principle for spatial spreading speeds and generalized propogating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), pp. 5125-5168. [47] W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), pp. 69-93. [48] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dis- persal evolution operators, J. Di erential Equations, 235 (2007), pp. 262-297. [49] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dis- persal in space periodic habitats, Journal of Di erential Equations 249 (2010), pp. 747-795. [50] W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Communications on Applied Nonlinear Analysis, 19 (2012), pp. 73-101. [51] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monos- table equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), pp. 1681-1696. [52] W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, submitted. [53] W. Shen and X. Xie, Approximations of random dispersal operators/equations by non- local dispersal operators/equations, submitted. [54] K. Uchiyama, The behavior of solutions of some nonlinear di usion equations for large time, J. Math. Kyoto Univ., 183 (1978), pp. 453-508. [55] H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), pp. 353-396. [56] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), pp. 511-548. 86 Appendices 87