TRAVELING FRONTS AND ENTIRE SOLUTIONS IN PARTIALLY DEGENERATE REACTION-DIFFUSION SYSTEMS WITH MONOSTABLE NONLINEARITY

This paper is concerned with traveling fronts and entire solutions for a class of monostable partially degenerate reaction-diffusion systems. It is known that the system admits traveling wave solutions. In this paper, we first prove the monotonicity and uniqueness of the traveling wave solutions, and the existence of spatially independent solutions. Combining traveling fronts with different speeds and a spatially independent solution, the existence and various qualitative features of entire solutions are then established by using comparison principle. As applications, we consider a reaction-diffusion model with a quiescent stage in population dynamics and a man-environment-man epidemic model in physiology.

1. Introduction.In the past decades, quite a few reaction-diffusion systems that some but not all diffusion coefficients are zeroes called partially degenerate reactiondiffusion systems, have been introduced to give an accurate description of a wide variety of phenomena in population biology, epidemiology, and so on.See the model u t (x, t) = du xx (x, t) − a 11 u(x, t) + a 12 v(x, t), v t (x, t) = −a 22 v(x, t) + g(u(x, t)), which is proposed by Capasso and Maddalena [1] to study the fecally-orally transmitted diseases in the European Mediterranean regions, and the following reactiondiffusion system in [11] u t (x, t) = du xx (x, t) + f 1 (u(x, t)) − γ 1 u(x, t) which describes a species population where the individuals alternate between mobile and stationary states, and only the mobile reproduce.For more details, we refer to [6,14,30,31] and the references cited therein.
Traveling wave solutions of partially degenerate reaction-diffusion systems, especially for the models (1) and (2) have been widely studied.For example, Xu and Zhao [28] proved the existence, uniqueness and stability of traveling fronts of (1) with bistable nonlinearity, and Zhao and Wang [32] established the existence of a minimal wave speed of (1) with monostable nonlinearity.For system (2), Zhang and 922 SHI-LIANG WU, YU-JUAN SUN AND SAN-YANG LIU Zhao [30] considered the asymptotic behavior of solutions, and Zhang and Li [31] further established the monotonicity and uniqueness of traveling wave solutions.Recently, Fang and Zhao [6] studied the traveling fronts and spreading speed of a general partially degenerate reaction-diffusion system.Li [15] further considered the traveling fronts for a class of partially degenerate reaction-diffusion systems that can have three or more equilibria.However, for the monostable case, these studies only considered the existence and non-existence of the traveling wave solutions.The first purpose of this paper is to further consider the asymptotic behavior, monotonicity and uniqueness of the monostable traveling wave solutions.
Although the study of traveling wave solutions is an important issue of reactiondiffusion equations, it is not enough for mathematical understanding of the dynamical structure of solutions.In fact, traveling wave solutions are only special examples of the so-called entire solutions that are defined for all time t ∈ R and for all point x ∈ R. Recently, there are quite a few works devoted to the entire solutions of scalar reaction-diffusion equations with and without delays, see e.g., [4, 5, 7-9, 12, 13, 16, 19, 24-26, 29].On the other hand, to the best of our knowledge, there were only four papers studying on the entire solutions of reactiondiffusion systems [10,20,22,27], where the existence of entire solutions for some specific reaction-diffusion model systems was established by employing comparison principle.The second purpose of this paper is to extend the works [10,20,22,27] to a large class of monostable partially degenerate reaction-diffusion systems.
More precisely, in this paper, we consider the traveling fronts and entire solutions of the following partially degenerate reaction-diffusion system u t (x, t) = du xx (x, t) + f (u(x, t), v(x, t)), v t (x, t) = −βv(x, t) + g(u(x, t)), which is a generalized version of the models (1) and (2).The nonlinearity of (3) is induced by the functions f and g, which satisfies the following conditions: (A 1 ): g ∈ C 2 ([0, K 1 ], R), g(0) = 0, K 2 = g(K 1 )/β > 0, f ∈ C 2 ([0, K 1 ] × [0, K 2 ], R), f (0, 0) = f (K 1 , K 2 ) = 0 and f u, 1 β g(u) > 0 for u ∈ (0, K 1 ), where K 1 is a positive constant; (A 2 ): g (u) > 0 for u ∈ [0, K 1 ], β∂ 1 f (K 1 , K 2 ) + g (K 1 )∂ 2 f (K 1 , K 2 ) < 0, and . We shall use a similar argument as in [10,20,22] to consider the entire solutions of (3).The idea is to study the solutions w n (x, t) = (u n (x, t), v n (x, t)) of Cauchy problems for (3) starting at times −n(n ∈ N) with appropriate initial conditions.By constructing appropriate sub-and supersolutions, some new entire solutions are obtained by passing the limit n → ∞.Although our method is similar to the works [10,20,22], the technique details are different.For example, for the partially degenerate system (3), the sequence functions v n (x, t) are not smooth enough with respect to x due to zero diffusion coefficient in v-equation, and hence its convergence is not ensured.To obtain a convergent subsequence, we have to make {v n (x, t)} possess a property which is similar to a global Lipschitz condition with respect to x (Lemma 15).
Throughout this paper, we always use the usual notations for the standard ordering in R 2 .We also use • to denote the Euclidean norm in R 2 .Now, we state our main results as follows.
Theorem 1. Assume (A 1 )-(A 3 ).Then, the following result holds: (ii) for each c ≥ c min , the traveling wave solutions of (3) with speed c are unique up to translations.
In the sequel, we always assume Φ c (•) = (φ c (•), ψ c (•)) is a traveling wave solution of (3) with speed c ≥ c min .To obtain the existence and qualitative properties of entire solutions other than the traveling fronts and the spatially independent solution, we need a stronger condition (A 3 ) as follows: Theorem 3. Assume (A 1 ), (A 2 ) and (A 3 ) .Then, for any θ 1 , θ 2 ∈ R and c 1 , c 2 ≥ c min , there exists an entire solution and Furthermore, the following statements hold: where Let Γ(t) be the spatially independent solution of (3) given in Theorem 2. We can consider any combination of traveling fronts and the spatially independent solution to construct some new entire solutions.For convenience, we define for w 1 = (u 1 , v 1 ) and w 2 = (u 2 , v 2 ).
Remark 1.When the solutions of (3) are assumed to range in [0, K], system (3) can be decoupled by solving the second equation and transformed into the scalar equation with infinite delay: In [16,23,24], Li et al. considered the traveling fronts and entire solutions of a nonlocal reaction-diffusion equation with finite delay of the form: where τ ≥ 0 is a finite constant.For the case c > c min , the monotonicity and uniqueness of the traveling wave solutions of (3) can be obtained from the arguments of Wang et al. [23].Thus, the new result in Theorem 1 is to guarantee such monotonicity and uniqueness when c = c min .
As far as the entire solution is concerned, the argument of [16,24] is similar to those of [4,9,12].More precisely, they studied the solutions u n (x, t)(n ∈ N) of Cauchy problems for (9) starting at times −n with appropriate initial conditions.By constructing appropriate subsolutions and supersolutions and establishing some priori estimates of solutions, the entire solutions are obtained by passing the limit n → ∞.However, for the equation with infinite delay, such as (8), a lack of regularizing effect occurs, see e.g., [21].Thus, the sequence functions {u n (x, t)} are not smooth enough, and hence its convergence is not ensured.Therefore, the existence and qualitative properties of the entire solutions of ( 8) and (3) (Theorems 3 and 4) can not be obtained directly from the results of [16,24].
The rest of this paper is organized as follows.In Section 2, we first transform the corresponding wave system into a scalar problem with an integral term.This property is effectively used to investigate the traveling wave solutions of (3).Then, the asymptotic behavior of the wave profiles at ±∞ is established by using the Ikehara's theorem [2].At the end of Section 2, we prove the monotonicity and uniqueness of the traveling wave solutions by using a sliding method, see e.g., Chen and Guo [3].Section 3 is devoted to the existence and asymptotic behavior of the spatially independent solution.In Section 4, Theorems 3 and 4 are proved by using comparison principle and constructing appropriate sub-super-solution pairs.The method is inspired by Guo and Morita [9] and Chen and Guo [4], see also Wang et al., [16,24].As applications, the main results are applied to the models (1) and (2) in Section 5.
2. Properties of traveling wave solutions.In this section, we study the asymptotic behavior, monotonicity and uniqueness of traveling wave solutions of (3).Throughout this section, we assume (A 1 )-(A 3 ).
Step 1.We show that φ(ξ) is integrable on (−∞, ξ ] for some ξ ∈ R. Step 2. We prove that φ(ξ) = O(e γξ ) as ξ → −∞ for some γ > 0. To get the assertion, we first show that Step 3.For 0 < Reλ < γ, define a two-sided Laplace transform of φ by Integrating the two sides of the equality (21) from −∞ to ξ, we obtain Then, it is easy to verify that for c > c min , lim Next, we prove ( 16) and (18).Since It follows from the second equation of (4) that lim for c > c min .Therefore, (16) holds.Similarly, one can show that (18) holds.This completes the proof.
Corollary 1.Let the assumptions of Theorem 5 be satisfied.Then, for all c ≥ c min , and 2.2.Monotonicity and uniqueness of traveling wave solutions.We first transform the monotonicity and uniqueness of traveling wave solutions of (3) to those of solutions of the scalar equation (11).
Lemma 5. Assume that (φ 1 , ψ 1 ) and (φ 2 , ψ 2 ) are traveling wave solutions of (3) with speed c ≥ c min .If there exists The proof is easy, so we omit it.From Lemmas 4 and 5, to prove the monotonicity and uniqueness of traveling waves solutions of system (3), it suffices to prove those of solutions of equation (11).Lemma 6. Assume that φ 1 and φ 2 are two solutions of ( 11) and (12) Proof.Suppose that there exists Let φ be a solution of (11) and (12) with c ≥ c min .Then φ (•) ≥ 0 on R. Proof.Due to Corollary 1, we know that φ is strictly increasing in R \ [−N, N ] for some N 1.By Lemma 6 and using the sliding method similar to that of [3, Lemma 4.3], one can easily show that φ (ξ) ≥ 0 for ξ ∈ R.This completes the proof.
The monotonicity of traveling wave solutions of (3), i.e, Theorem 1(i), is a direct consequence of Lemmas 4 and 7.
The following two lemmas are important to prove the uniqueness of solutions of (11).
Lemma 10.Assume that φ 1 and φ 2 are two solutions of ( 11) and ( 12) with c ≥ c min .Then there exists Proof.By virtue of Lemmas 6 and 9 and applying the sliding method, the proof is similar to that of [3,Theorem 5.1].We omit it here.
The uniqueness of traveling wave solutions of (3), i.e., Theorem 1(ii), is a direct consequence of Lemmas 5 and 10.
We now consider the spaces C(R, R) of continuous real functions on R, and the operator T : It is easy to see that the following result holds.
Direct computations show that the following result holds.
Remark 2. (i) It should be point out the existence of Γ(t) can also be established using the monotone dynamical systems theory.However, the exponential decay rate of the spatially independent solution at minus infinity can not be obtained using this method.
(ii) To obtain the existence of the monotone spatially independent solution, the condition g (u) > 0 for u ∈ [0, K 1 ] and 4. Existence of entire solutions.In this section, we first state some definitions and comparison theorems, and give a priori estimate on solutions of (3).Then we prove the main results of entire solutions by using the comparison argument and constructing appropriate sub-super-solution pairs.Throughout this section, we always assume (A 1 ), (A 2 ), and (A 3 ) .4.1.Preliminaries.Let X = BUC(R, R 2 ) be the Banach space of all bounded and uniformly continuous functions from R into R 2 with the supremum norm Let T 2 (t)ψ = e −βt ψ, ∀ψ ∈ BUC(R, R) and T 1 (t) be the analytic semigroup on BUC(R, R) generated by u t = du xx .Clearly, T (t) = (T 1 (t), T 2 (t)) is a linear semigroup on X.
Proof.From Lemma 14, we see that 0 ≤ (u(x, t), v(x, t)) ≤ K for all x ∈ R and t ≥ 0. By the second equation of (3), we have for x ∈ R and t ≥ 0, Note that for any s ≥ 0 and t > s, Consequently, for any s ≥ 0 and t ∈ [s + 1, s + 4], 4d(t−s) u(y, s)dy Similar to the proof of [24, Proposition 4.3], we can easily verify that where Since s ≥ 0 is arbitrary, which implies that Using the estimate for v t and applying a similar argument, we can find a positive constant M 3 , which is independent of ϕ, such that for any x ∈ R and t > 1, Then, for any x ∈ R and t > 1, Then the first statement of this lemma follows.Now we prove (29).Note that v(x, t) = ϕ 2 (x)e −βt + t 0 g u(x, s) e −β(t−s) ds, ∀x ∈ R, t > 0.
By our assumption, we have for any η > 0, x ∈ R and t > 1, Moreover, for any η > 0, x ∈ R and t > 1, Here holds.This completes the proof.
), for x ∈ R and t > 0. Then w + (x, t) ≥ w − (x, t) for all x ∈ R and t ≥ 0.
Proof.Set w(x, t) = w 1 (x, t), w 2 (x, t) := w + (x, t) − w − (x, t) for x ∈ R and t ≥ 0, then w(x, t) satisfies w(x, 0) ≥ 0 and for x ∈ R and t > 0. From (30), we get for all x ∈ R and t ≥ 0, where Using the same method as that in [25,Theorem 3.4], it is easy to prove that w 2 (x, t) ≥ 0 for all x ∈ R and t ≥ 0. It then follows from (32) that w 1 (x, t) ≥ 0 for all x ∈ R and t ≥ 0. Therefore, w + (x, t) ≥ w − (x, t) for x ∈ R and t ≥ 0. This completes the proof.
In order to construct supersolutions of (3), we make the following extension for the function g.Let σ > 1 be a constant.Define ĝ : [0, For convenience, we denote ĝ by g in the remainder of this paper.
Proof.We only consider the case c 2 ≥ c 1 > c min , since the other cases can be discussed similarly.For convenience, we denote Define The rest of the proof is divided into three steps.
Step 1.We verify that where We claim that ), then we have Thus, we have Similarly, we can show that Then, the claim follows.
Step 3. We now show that there exists T < 0 such that w(x, t) is a supersolution of (3) on (−∞, T ).We first show the following claim.Claim.There exists T 1 < 0 such that for every t < T 1 , there are only a finite number of points in x ∈ R so that φ c1 (x + p 1 (t)) + φ c2 (−x + p 2 (t)) = K 1 .
Then the claim follows.Similarly, we can show that there exists T 2 < 0 such that for every t < T 2 , there are only a finite number of points in x ∈ R so that ψ c1 (x + p 1 (t)) + ψ c2 (−x + p 2 (t)) = K 2 .
Proof of Theorem 3.For n ∈ N, we denote Consider the following initial value problem of (3): We first show the following claim.Claim.ϕ n ∈ [0, K] X and there exists L > 0, which is independent of n, such that Clearly, 0 ≤ ϕ n (•) ≤ K.Note that 0 ≤ ψ c (t) ≤ 1 c [βK 2 + g(K 1 )] := L c , ∀t ∈ R, and for each n ∈ N, there exists x n ∈ R such that x < x n .
For any x ∈ R and η > 0, if x + η > x ≥ x n , then where λ 3 , λ 4 , L 1 , H(φ) are defined as in the proof of Lemma 3, we can similarly show that for any η > 0, Therefore, we obtain ϕ n ∈ [0, K] X and the claim follows.
We now prove (vi).Assume It then follows from Lemma 16 that 0 ≤ Z(x, t) ≤ V (x, t) for x ∈ R and t > −n, that is In view of lim n→+∞ w n (x, t) = W c1,c2,ω1,ω2 (x, t), we get for all (x, t) ∈ R 2 , which implies that W c1,c2,ω1,ω2 (x, t) converges to Φ c1 (x+c Finally, we show that (vii) and (viii) hold.Note that for any fixed x ∈ R and t −1, It then follows from Theorem 5 and d dc cλ 1 (c) < 0 for c > 0 that (vii) holds.Using the assertion (vii) and Lemmas 1 and 3, the proof of (viii) is similar to that of [16,Theorem 1.1(vii)] and omitted.
This completes the proof of Theorem 3.
The proof is similar to that of Lemma 17, see also [16,Lemma 3.6].We omit it here.
Proof.We only consider the case The remainder of the proof is divided into three steps.
Step 3. By a similar argument as Lemma 18, we can prove that there exists T < 0 such that w + (x, t) is a supersolution of (3) on (−∞, T ].The proof is complete.

TRAVELING FRONTS AND ENTIRE SOLUTIONS 945
Proof of Theorem 4. The proof of Theorem 4 is similar to that of Theorem 3 according to Lemmas 19 and 20, here we omit it.