Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It first introduces the concept of generalized traveling wave solutions of time recurrent and space periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It then proves that in the direction of any unit vector ξ, there is c∗(ξ) such that for any c > c∗(ξ), a generalized traveling wave solution in the direction of ξ with averaged propagation speed c exists. It also proves that if the time recurrent and space periodic monostable equation is indeed time periodic, then c∗(ξ) is the minimal wave speed in the direction of ξ and the generalized traveling wave solution in the direction of ξ with averaged speed c > c∗(ξ) is a periodic traveling wave solution with speed c, which recovers the existing results on the existence of periodic traveling wave solutions in the direction of ξ with speed greater than the minimal speed in that direction.


Introduction
The current paper is devoted to the study of traveling wave solutions of reaction diffusion equations of the form, where a i (t, x) (i = 1, 2, · · · , N ) and f (t, x, u) are recurrent and unique ergodic in t and periodic in x, and f (t, x, u) is monostable in u. More precisely, a i (t, x) (i = 1, 2, · · · , N ) and f (t, x, u) satisfy the following three assumptions.
x, u), ∂f ∂u (t, x, u)) is recurrent and unique ergodic in t (see Definition 2.2 for detail), is periodic in x j with period p j (j = 1, 2, · · · , N ), and is globally Hölder continuous in t, x.
(H2) There are β 0 > 0 and P 0 > 0 such that f (t, u) ≤ −β 0 for t ∈ R and u ≥ P 0 and ∂f ∂u (t, u) ≤ −β 0 for t ∈ R and u ≥ 0. (H3) The principal Lyapunov exponent of the linearization of (1.1) at 0 is positive (see section 2.2 for the definition of principal Lyapunov exponent and basic properties).
Observe that under the assumptions (H1)-(H3), the trivial solution u ≡ 0 of (1.1) is linearly unstable and (1.1) has a unique positive solution u = u + (t, x) which is recurrent and unique ergodic in t, periodic in x i with period p i , and is globally stable with respect to positive space periodic perturbations (see Proposition 3.1).
Equation (1.1) satisfying (H1)-(H3) is hence called a monostable equation. Here is a typical example of monostable equations, which was introduced in the pioneering papers of Fisher [9] and Kolmogorov, Petrowsky, Piscunov [23] for the evolutionary take-over of a habitat by a fitter genotype, where u is the frequency of one of two forms of a gene. Monostable equations are then also called Fisher's or KPP type equations in literature. They are used to model many other systems in biology and ecology (see [1], [2], [5]). One of the central problems about monostable equations is the traveling wave problem. This problem is well understood for the classical Fisher or KPP equation (1.2). For example, Fisher in [9] found traveling wave solutions u(t, x) = φ(x − ct), (φ(−∞) = 1, φ(∞) = 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 [23] by Kolmogorov, Petrowsky, and Piscunov, that is, they proved that for any nonnegative solution u(t, x) of (1.2), if at time t = 0, u is 1 near −∞ and 0 near ∞, then lim t→∞ u(t, ct) is 0 if c > 2 and 1 if c < 2. Put c * = 2. c * is of the following spatially spreading property: for any nonnegative solution u(t, x) of (1.2), if at time t = 0, u(0, x) ≥ σ for some σ > 0 and x −1 and u(0, x) = 0 for x 1, then inf x≤c t |u(t, x) − 1| → 0, ∀c < c * and sup x≥c t u(t, x) → 0 ∀c > c * as t → ∞.
Recall that when (1.1) is time periodic in t with period T , it has been proved that for any ξ ∈ R N with ξ = 1, there is a c * (ξ) ∈ R such that for any c ≥ c * (ξ), there is a traveling wave solution connecting u + and 0 and propagating in the direction of ξ with speed c, and there is no such traveling wave solution of slower speed. The minimal wave speed c * (ξ) is of some important spreading properties and is called the spreading speed in the direction of ξ (see [26], [31], [33], [47], and references therein). Moreover, the following variational principle for c * (ξ) holds, where λ * (µ, ξ) is the principal eigenvalue (i.e. the eigenvalue with largest real part and a positive eigenfunction) of the following periodic parabolic eigenvalue problem, However, there is little understanding of the traveling wave problem for general time dependent and space periodic monostable equations. The objective of the current paper is to investigate the extent to which the concepts and theories of traveling wave solutions of time and/or space periodic monostable stable equations may be generalized.
To this end, we first introduce the concept of generalized traveling wave solutions, which generalize the classical concept of traveling wave solutions. Roughly, a solution u = u(t, x) of (1.1) is called a generalized traveling wave solution of average propagating speed c in the direction of ξ ∈ S N −1 if it is an entire solution of (1.1) and is a generalized traveling wave solution of (1.1) in the direction of ξ, then it is also of the following form, To state the main results of the paper, for given ξ ∈ S N −1 and µ > 0, let λ(µ, ξ) be the principal Lyapunov exponent (see Definition 2.4) of x, 0). Observe that principal Lyapunov exponent of (1.7) is the analogue of the principal eigenvalue of the periodic parabolic problem (1.4) (see section 2 for basic properties of principal Lyapunov exponents of time recurrent parabolic equations). When a i (t, x) and a 0 (t, x) are periodic in t with period T , λ(µ, ξ) equals the principal eigenvalue λ * (µ, ξ) of (1.4).
As mentioned above, if a i (t, x) and f (t, x, u) are periodic in T , it has been proved (see [31], [47]) that for any c ≥ c * (ξ), (1.1) has a periodic traveling wave solution in the direction of ξ. The results of the current paper recover the existing results on periodic traveling wave solutions with speed c > c * (ξ). It should be pointed out that the approach used in this paper is different from the approaches in other papers.
When (1.1) is recurrent and unique ergodic in t but independent of x, it has been proved in [43] that for any c ≥ c * (ξ), (1.1) has a generalized traveling wave solution in the direction of ξ with averaged speed c. Hence the results of the current paper also recover the existing results on generalized traveling wave solutions with average propagating speed c > c * (ξ).
Observe that when (1.1) is periodic in t or is independent of x, it has been proved that c * (ξ) is the spreading speed as well as the minimal average propagating speed of generalized traveling wave solutions of (1.1) in the direction of ξ (see [42], [43], [47]) (the reader is referred to [15], [42] for the studies of spreading speeds of time recurrent monostable equations). However it remains open whether in general c * (ξ) is the spreading speed of (1.1) in the direction of ξ (if exists) and is the minimal average propagating speed of generalized traveling wave solutions in the direction of ξ.
The rest of the paper is organized as follows. In section 2, we collect some fundamental properties of principal Lyapunov exponent of (1.7), which together with comparison principles for parabolic equations are the main tools for the proofs of the main results of the paper. We also recall the concepts of recurrent and unique ergodic functions and compact minimal flows in section 2. We prove the existence, uniqueness, and stability of space periodic and time recurrent and unique ergodic positive solutions of (1.1) and construct some important super-and sub-solutions of (1.1) in section 3. In section 4, we study the existence of generalized traveling wave solutions of (1.1) and prove the main results.

Preliminary
In this section, we first recall the concepts of compact minimal flows and recurrent and unique ergodic functions. We then collect some fundamental properties of principal Lyapunov exponent of (1.7).

Compact flows and recurrent functions
In this subsection, we recall the definitions of compact flows and recurrent functions and collect some basic properties.
is jointly continuous in (t, z) ∈ R×Z, σ 0 = id, and σ s •σ t = σ s+t for any s, t ∈ R. We may write z · t or (z, t) for σ t z.
(2) Assume that (Z, {σ t } t∈R ) is a compact flow. A probability measure µ on ergodic if it has a unique invariant measure (in such case, the unique invariant measure is necessarily ergodic).  [40]).
is periodic or almost periodic in the first independent variable and is bounded and uniformly continuous on R × E for any bounded subset E ⊂ R n , then it is both recurrent and unique ergodic in the first variable. The reader is referred to [8] for the definition and basic properties of almost periodic functions.
is recurrent and unique ergodic in the first independent variable, then the limit lim t−s→∞ Proof. It follows from the results contained in [21].

Spectral theory for linear recurrent parabolic equations with periodic boundary conditions
In this subsection, we collect some fundamental properties of the principal spectrum for time recurrent parabolic equations with periodic boundary conditions. The reader is referred to [42] for detail. The reader is also referred to [17], [18], [19], [20], [28], [29], [37], [42] for the studies of principal spectral theory for general time dependent parabolic equations with Dirichlet, or Neumann, or Robin boundary conditions. We first consider a family of linear parabolic equations of the form, complemented with the periodic boundary condition To emphasis the dependence of (2.1) on b, we may write it as (2.1) b .
We make the following standard assumption on Y .
is recurrent in t and periodic in x j with period p j > 0 (j = 1, 2, · · · , N ) and are globally Hölder continuous in t, and is connected and compact under open compact topology.
In the following, we assume that Y satisfies (H-Y) and Y is equipped with the open compact topology. Then (Y, (σ t ) t∈R ) is a compact flow. For a given Banach space X, · X denotes the norm in X. Let equipped with uniform convergence topology. Let L(X L , X L ) be the space of bounded linear operators from X L to X L . Let and and It follows from [14] that −∆ is a sectorial operator on Then by [14], for any b ∈ Y and u 0 ∈ X α0 L , there is a unique solution Following from the results in [14] and classical theory for parabolic equations, we have By Theorem 2.2, (2.1)+(2.2) generates a skew-product semiflow on X α0 L × Y : The following Theorem follows from [38] (see also [28], [29]).
and satisfy the following properties: (2) X α0,1 By classical theory for parabolic equations and the continuity ofw(b) in b ∈ Y with respect to the · α0 -norm, there are constants C 1 , C 2 > 0 such that and ·, · be the inner product in L 2 (D). Let Therefore Proof. It follows from Theorems 2.3, 2.4, and the results in [21].
We now consider a single linear parabolic equation complemented with the periodic boundary condition (2.2), where ({a i (t, x)} N i=1 , a 0 (t, x)) is recurrent and unique ergodic in t and periodic in x j with period p j (j = 1, 2, · · · , N ), and are globally Hölder continuous in t, x.
Observe that λ(µ, ξ; a) and w µ,ξ (t, ·; a) are analogs of principal eigenvalues and principal eigenfunctions of elliptic and periodic parabolic problems, respectively. In literature, {span w(σ t a µ,ξ ) } t∈R is call the principal Floquet bundle of (2.10) associated to the principal Lyapunov exponent.
Observe that for any u 0 ∈ C(R N , R) with for some α, C > 0, (2.9) has also a unique solution u(t, x; s, u 0 , a) with u(s, x; s, u 0 , a) = u 0 (x) (see [11]). Regarding solutions of (2.9) without the periodic boundary condition, we have Theorem 2.7. For any given ξ ∈ S N −1 and µ > 0, is a solution of (2.9).
When a i and a 0 are periodic in t with period T , let λ * (µ, ξ; a) be the principal eigenvalue of the periodic parabolic eigenvalue problem (1.4) and w * (t, x; µ, ξ, a) be an associated positive principal eigenfunction. Then Theorem 2.8. For any given ξ ∈ S N −1 , µ > 0, is a solution of (2.9).

Positive recurrent solutions and super-, sub-solutions
In this section, we first prove the existence, uniqueness, and stability of space periodic and time recurrent positive solutions of (1.1). Then we construct some important super-and sub-solutions of (1.1) to be used in the proofs of the main results in next section. Throughout this section, we assume (H1)-(H3).
For any z ∈ R N , consider also equipped with uniform convergence topology. For given u 1 , u 2 ∈ X, we write Let X L and X + L be as in (2.3) and (2.4), respectively. By the classical theory for parabolic equations (see [11], [14]), for any u 0 ∈ X, (3.1) has a unique (local) solution u(t, ·; u 0 , b, g) with initial condition u(0, ·; u 0 , b, g) = u 0 (·). A function u(t, x) is called an entire solution of (3.1) if it is a solution of (3.1) for t ∈ R and x ∈ R N . Note that if u 0 ∈ X + , then u(t, ·; u 0 , b, g) exists for all t > 0 and if u 0 ∈ X L , then u(t, ·; u 0 , b, g) ∈ X L for t ≥ 0 at which u(t, ·; u 0 , b, g) exists.

Proposition 3.1 (Positive recurrent and unique ergodic solution).
There is a continuous function q : H(a, f ) → X + L \ {0} such that u(t, ·; q(b, g), b, g) = q((b, g) · t) and u(t, ·; q(b, g), b, g) is globally stable in the sense that for any , x; q(a, f ), a, f ). Then u + (t, x) is recurrent and unique ergodic in t.
In the following, we write u(t, x; q(a, f ), a, f ) as u + (t, x; a, f ) or simply as u + (t, x) if no confusion occurs. Let Observe that u + inf > 0. Similarly, for any u 0 ∈ X, s ∈ R, and z ∈ R N , (3.2) has a unique (local) solution u(t, ·; s, z, u 0 ) with u(s, x; s, z, u 0 ) = u 0 (x). If u 0 ∈ X + , u(t, ·; s, z, u 0 ) exists for all t > s. The following proposition follows easily.
Proposition 3.2. Assume that u n , u 0 ∈ X, u n ≤ M for n = 1, 2, · · · and some M > 0, and u n (x) → u 0 (x) as n → ∞ uniformly for x in bounded subsets of R N . Then for any t > 0, u(t, x; s, z, u n ) → u(t, x; s, z, u 0 ) as n → ∞ uniformly for s ∈ R and x in bounded subsets of R N .
Proof. (i) It follows from a direct calculation.
(ii) For any d, d 1 with 0 < d 2 ≤ d 1 and d < d 0 and any u 0 ∈ X + with u 0 (x) ≥ max{ψ 2 (s, x; s, z), 0}, Proof. The proposition can be proved by the arguments similar to those in [43,Lemma 3.3]. For the completeness and the reader's convenience, we provide a proof in the following.
(i) First of all, it is clear that for d sufficiently small, we have ψ 2 (t, x; s, z) ≤ u + inf (≤ u + sup + 1) for t ≥ s and x ∈ R N , where u + sup and u + inf are as in (3.4) and (3.5), respectively.

Generalized traveling wave solutions
In this section, we investigate the existence of generalized traveling wave solutions of (1.1). First, we introduce the notion of generalized wave solutions. (1) An entire solution u(t, x) of (1.1) is called a generalized traveling wave solution of (1.1) in the direction of ξ (connecting u + (·) and 0) with average propagating speed c (or averaged wave speed c) if there are ζ : R → R (ζ(0) = 0) and Φ : (2) We say that Φ : R N × R × R N → R + generates a generalized traveling wave solution of (1.1) in the direction of ξ with average propagating speed c if there is ζ : is an entire solution of (1.1).
uniformly in t ∈ R and z ∈ R N .
The main results of the paper state as follows.
(3) If a i (t, x) (i = 1, 2, · · · , N ) and f (t, x, u) are periodic in t with period q, then is a periodic traveling wave solution of (1.1).
Proof. First of all, let d 1 = d 2 and d < d 0 , where d 0 is as in Proposition 3.5. For any given s < t, let For each m ∈ N, s ∈ R, and z ∈ R N , let Then u − m,s,z (·) ≤ u + m,s,z (·). By Propositions 3.4 and 3.5, Observe that η m µ1 (s, x; −mT + s, z) = w(σ s a µ1,ξ )(z + x). This together with (4.5) implies that Note that for any r > 0, This together with (4.7) implies that for m = 1, 2, · · · . Hence there are u * s,z (x) ≥ 0 such that lim m→∞ u m,s,z = u * s,z (x), uniformly for x in bounded subsets of R N . By (4.5) for x ∈ R N and s ∈ R. By (4.8), . We prove that Φ(·, ·, ·) satisfies the conclusions in the theorem.
(3) It follows from the construction.