Traveling Wave Solutions of a Reaction-Diffusion Equation with State-Dependent Delay

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem.


Introduction
Much research efforts have been devoted to understanding the dynamics of differential equations with state-dependent delay.Differential equations with state-dependent delay are useful to study population dynamics in which the amount of food available per biomass for a fixed food supply is a function of the total consumer biomass [1].Andrewartha and Birch [2, pp. 370] studied a differential equation with state-dependent in which the duration of larval development of flies is a nonlinear increasing function of larval density.
There is a wealth of literature on the research and we refer to Arino et al. [3], Cooke and Huang [4], Hartung et al. [6], Hu et al. [7], Magal and Arino [12], Mallet-Paret and Nussbaum [13], Walther [17] and the references cited therein.Despite that the dynamics of functional differential equations has been widely studied, the spatial-temporal patterns of differential equation with state-dependent delay are hardly understood.
In this paper, we consider the existence and nonexistence of traveling wave solutions of the following reaction-diffusion equation with state-dependent delay where x ∈ R, t > 0. In population dynamics, d > 0 is the death rate and b : [0, ∞) → [0, ∞) is the birth function satisfying (B) b(0) = 0, b(K) = K for some positive constant K > 0, and b(u) > du, u ∈ (0, K) while 0 < b(u) < du, u > K, which will be imposed throughout this paper.In this model, time delay is not a constant and τ : [0, ∞) → [0, ∞) satisfies the following assumptions: To continue our discussion, we first present the following definition.
Definition 1.1 A traveling wave solution of (1.1) is a special entire solution defined for all x, t ∈ R and taking the following form where c > 0 is the wave speed and φ ∈ C 2 (R, R) is the wave profile.In particular, if φ(ξ) is monotone increasing, then it is a traveling wavefront.
By the above definition, φ and c must satisfy the following functional differential equation of second order In particular, to model precise transition processes in evolutionary systems by traveling wave solutions, we also consider the following asymptotic boundary conditions or a weaker version When τ ′ (u) = 0, the traveling wave solutions of (1.1) have been widely studied since Schaaf [14].For monotone b(u) with τ ′ (u) = 0, comparison principle is admissible, the existence and nonexistence of monotone traveling wave solutions can be studied by monotone iteration, fixed point theorem and monotone semiflows, see Liang and Zhao [8], Ma [10], Wu and Zou [19].If b(u) is locally monotone and τ ′ (u) = 0 holds, then the traveling wave solutions of (1.1) can be studied by constructing auxiliary monotone equations, see Fang and Zhao [5], Ma [11], Wang [18].
However, when τ (u) is not a constant, because τ (u) is nondecreasing, the comparison principle in (1.1) does not hold even if b(u) is monotone increasing.Therefore, the study of traveling wave solutions of (1.1) needs some new techniques.In this paper, similar to that in Wu and Zou [19], we first introduce an integral operator to study the existence of we obtain an auxiliary equation with fixed time delay by (1.4).Applying the theory of asymptotic spreading, we establish the nonexistence of traveling wave solutions.

Some Estimations
Let β > d hold and define H : Further define an operator F : where Clearly, a fixed point of F is a solution of (1.2), and a solution of (1.2) is a fixed point of F. Therefore, to study the existence of solutions of (1.2), it suffices to investigate the existence of fixed points of F.
Lemma 2.2 Assume that c > 0 is fixed and Proof.By the definition of F, we have The proof is complete.
Then the result is clear by the definition of β.The proof is complete.
From the monotonicity of b, we further have Then the result is clear by the definition of β.The proof is complete.

Upper and Lower Solutions
To investigate the existence of (1.2)-(1.3),we will use the upper and lower solutions defined as follows.
A lower solution can be similarly defined by inversing the inequality.
To construct upper and lower solutions, we define some constants.For λ ≥ 0, c ≥ 0, we first define Lemma 2.6 There exists c * > 0 such that Λ(λ, c) > Proof.Evidently, we have Then the result is clear and we complete the proof.
We now consider the existence of traveling wave solutions if c > c * is a constant and define continuous functions as follows where Lemma 2.8 For any c > c * , there exists Then the proof is complete.Proof.
> 0 by the definition of q.The proof is complete.

Existence of Monotone Traveling Wave Solutions: c > c *
In this part, we assume that c > c * is fixed and prove the existence of fixed points of F by Schauder's fixed point theorem.Moreover, β > 0 is a fixed constant satisfying Remark 2.9.Let µ ∈ (0, −γ 1 (c)) be a constant and Then it is easy to show that .
By what we have done, we see that Γ is nonempty and convex.Moreover, we can verify that Γ is closed and bounded with respect to the norm | • | µ .
Lemma 2.11 F admits the following nice properties.
Proof.From Lemma 2.11, it suffices to verify that By the definition of upper solution, we see that Then the continuity implies that In a similar way, we can prove that The proof is complete.
and so Applying these estimations, we have which implies the continuity in the sense of | • | µ .
Due to Lemma 2.1, it is easy to prove the compactness and we omit the details.The proof is complete.
We now present the main conclusion of this part.Then it is easy to verify that lim ξ→+∞ φ(ξ) = K.The proof is complete.

Existence of Monotone Traveling Wavefronts
In this part, we shall establish the existence of monotone solutions of (1.
We first present the main results as follows.Proof.Let {c n } n∈N be a decreasing sequence satisfying Then Theorem 2.14 implies that for each c n , F with c = c n has a fixed point φ n (ξ), where φ n (ξ) is monotone increasing.Because a traveling wavefront is invariant in the sense of phase shift, we assume that From Lemma 2.2, φ n (ξ) is equicontinuous and uniformly bounded for all n ∈ N, ξ ∈ R.
By Ascoli-Arzela Lemma and a standard nested subsequence argument, it follows that there exists a subsequence of {c n } n∈N , still denoted it by {c n } n∈N , such that the corresponding φ n (ξ) converges uniformly on every bounded interval of ξ ∈ R, and hence pointwise for ξ ∈ R to a function φ * (ξ).Moreover, it is evident that and the convergence is uniform in ξ, s ∈ R.
Let n → ∞ in F, then the dominated convergence theorem implies that Then the uniform continuity and the dominated convergence theorem in F lead to and so dφ ± = b(φ ± ).Again by (2.5), we have The proof is complete.In this section, we investigate the nonexistence of traveling wavefronts if c < c * .We first consider the following initial value problem . By Smith and Zhao [15], Thieme and Zhao [16], we have the following three lemmas.
) has a unique mild solution u(x, t) defined for all x ∈ R, t > 0.Moreover, u(x, t) is continuous in x ∈ R, t > 0, and satisfies Lemma 2.17 Assume that and satisfies ) Lemma 2.18 Assume that c > 0 such that 2) with (1.3) does not have a monotone solution.
From (2.9), u(x, t) = φ(x + c 1 t) satisfies which implies a contradiction because of The proof is complete.

Nonmonotone Birth Function
In this section, we investigate the existence and nonexistence of positive traveling wave solutions of (1.1), namely, existence and nonexistence of positive solutions of (1.2) and (1.4), if b : [0, ∞) → [0, ∞) satisfies the following assumptions: Clearly, if b(u) = pue −u , then (C1)-(C3) hold when p > d.In particular, (C1)-(C3) will be imposed throughout this section without further illustration.For convenience, we define We first present the nonexistence of traveling wave solutions.Then there exists Similar to the proof of Theorem 2.19, we can verify the result.
Clearly, both b(u) and b(u) are monotone and continuous and there exists k > 0 such that Consider the following two equations and in which all the parameters are the same as those in (1.1).Then the existence and nonexistence of monotone traveling wavefronts of (3.1) and (3.2) can be answered by the conclusions in Section 2.
Proof.For any fixed c > c * , we define where λ 1 (c) is the same as that in Section 2, ϕ(ξ) is a monotone traveling wavefront of (3.2) and satisfies By the discussion in Section 2, we see that Similar to the discussion in Section 2, we can select β, µ such that (P1) Γ * is nonempty, convex, bounded and closed; Then (3.5) is true.Because of φ * (0) < 3φ − 4 , it is evident that φ * (ξ) is not a constant.The proof is complete.
Remark 3.5 Similar to those in [5,11,18], we can obtain some sufficient conditions such that lim ξ→∞ φ(ξ) exists even if b is not monotone.
Unfortunately, we cannot obtain the existence of lim ξ→−∞ φ(ξ) in Theorem 3.4.However, under some additional assumptions, we can formulate the limit behavior as follows.
traveling wave solutions.Based on some estimations, we confirm the comparison principle on a proper subset of the space of continuous functions if b(u) is monotone.Then the existence of (1.2)-(1.3) is proved by combining Schauder's fixed point theorem with upper and lower solutions if the wave speed is larger than a threshold c * defined later.When the wave speed is c * , we establish the existence of (1.2)-(1.3)by passing to a limit.If the wave speed is less than c * , the nonexistence of (1.2)-(1.3) is confirmed.Therefore, c * is the minimal wave speed of (1.2)-(1.3).If b(u) is not monotone, similar to Ma[11], we introduce two auxiliary monotone equations to confirm the existence of (1.2) with (1.4) for c > c * .If c = c * , the existence of nontrivial solutions of (1.2) is also proved by passing to a limit function.When c < c * ,