Traveling wave solutions for a class of reaction-diffusion system

where β is a positive constant, u and v represent the concentration of ferritin and acid nitrate, respectively, and δ is the ratio of diffusion rate. Fu [2, 3] showed the existence and uniqueness of traveling solution for the acid nitrate-ferritin reaction model by using the perturbation method. In (1.1) the U(1 – U) is the logistic term, which means the birth rate minus death rate of U . There are also many scientists who study the existence, uniqueness, and stability of traveling wave solutions for a reaction-diffusion model in population biology and chemistry. We first recall some existing methods on the existence of traveling waves for the reaction-diffusion model. In [9], Trofimchuk, Pinto, and Trofimchuk studied the traveling wavefronts for a model of the Belousov–Zhabotinskii reaction in a chemical model by


Introduction
This paper deals with the existence and uniqueness of traveling wave solutions of the following reaction-diffusion system: (1.1) where r and b are positive constant numbers. In relation to our topic, Fu [2,3] studied the acid nitrate-ferritin reaction model as follows: where β is a positive constant, u and v represent the concentration of ferritin and acid nitrate, respectively, and δ is the ratio of diffusion rate. Fu [2,3] showed the existence and uniqueness of traveling solution for the acid nitrate-ferritin reaction model by using the perturbation method. In (1.1) the U(1 -U) is the logistic term, which means the birth rate minus death rate of U.
There are also many scientists who study the existence, uniqueness, and stability of traveling wave solutions for a reaction-diffusion model in population biology and chemistry. We first recall some existing methods on the existence of traveling waves for the reaction-diffusion model. In [9], Trofimchuk, Pinto, and Trofimchuk studied the traveling wavefronts for a model of the Belousov-Zhabotinskii reaction in a chemical model by constructing upper and lower method solutions. They showed the existence and uniqueness of solution. In [4], Huang investigated the existence of traveling wave for a class of predator-prey systems via the perturbation method. For more details on the existence of traveling wave solutions for other types of diffusion-reaction models, readers can refer to [5][6][7][8][10][11][12][13] and the references in these papers.
The remaining part of this paper is organized as follows. In Sect. 2, we construct the supersolution and subsolution of (2.1) and introduce some useful lemmas for the main result. In Sect. 3, we show the existence and uniqueness of traveling wave for (2.1).

Preparation
In this section, we introduce some useful lemmas for the main results of our papers. Note that some reaction-diffusion models in population biology can be rewritten as the form (1.1), and the existence and uniqueness of traveling waves for these reaction-diffusion equations are equivalent to those for (2.1). Throughout this paper, a traveling wave solution of (1.1) always refers to a pair (U, V , c), where U and V are bounded, continuous, nonnegative, and nonconstant functions from R to R such that u(t, x) := U(xct), v(t, x) := V (xct) satisfies (2.1). Clearly, U(z), V (z) satisfy the following wave profile system: There are boundary conditions (U, V )(-∞) = (0, 1/r) and (U, V )(+∞) = (1, 0). The main purpose of this paper is to study the existence and uniqueness (up to translation) of traveling waves for (2.1). Next, we construct the supersolution and subsolution of (2.1) that will be used in the following sections. For simplicity, we denote Due to c > c * , where c * is the minimum wave speed (2.1) and equation p(s) = 0 has two positive roots λ and λ + d, there are In addition, p(s) < 0, when s ∈ (λ, λ + d).
Proof When z < z 0 , it can be concluded from the above inequality that (-∞, z 0 ) is estab- By calculating (2.2), (2.4), and M > 1, we know Thus, we can conclude that (2.3) holds. The proof is completed. .
Proof If z < z 1 , from (2.3), we can get that there is V -≡ 0 on (-∞, z 1 ). If z > z 1 , there are V -= V + -Ke -(λ+η)z and U -= 1 -Me -γ z , and by calculating it is easy to see that We know .
The proof is completed.
Next, we establish some prior estimates of solutions to nonhomogeneous equations We need the following lemmas (Lemma 2.4 , Lemma 2.5, Lemma 2.6) (see [3]). For the convenience of readers, we give the details.

Lemma 2.4 Let A be a constant number and g is a continuous function on [a, b]. Let φ 1 and φ 2 be the solution of second-order linear equation L[y]
Then we have the following estimates of φ 1 and φ 2 : Proof If g ≤ 0, the Langsky matrix W (φ 1 , φ 2 ) of φ 1 and φ 2 can be estimated as In order to prove (2.6) and (2.7), rewrite the equation L[y] = 0 as a first-order system where · represents the absolute norm, because of Replacing a with b in (2.10), we get In order to estimate φ 1 (b), we make the second-order equation According to the maximum principles, we get ψ ≥ 0, so φ 1 ≥ ρ on [a, b], then Combining (2.13) and (2.14), we know Therefore, this lemma is proved.

Lemma 2.5 Let A be a positive constant, and let g and h be continuous functions on
There exist a positive constant K 1 that depends only on the length of [a, b] and a positive constant Since w, g, and h are right continuous at point a, thus w (a+) exists. Similarly, we also have the existence of w (b-). Let φ 1 and φ 2 be as shown in Lemma 2.4. For any z ∈ (a, b), Since w(a) = w(b) = 0, multiplying (2.15) and (2.16) by φ 1 , φ 2 respectively, we get Thus Since |h| ≤ K 2 , this implies Finally, by virtue of (2.6), (2.7), (2.8), and the above inequality, we can get that w C[a,b] ≤ K 2 K 3 holds.  ∈ (a, b), multiplying (2.15), (2.16) by φ 1 and φ 2 , respectively, we have It can be concluded that Under the assumptions of |h| ≤ K 2 and |w| ≤ K 0 , we know By virtue of (2.6), (2.7), and (2.8), we find that (2.17) holds. Next, we consider the existence and uniqueness of solution to problem (2.1) within the interval [-l, l]. The system is (2.18) By studying some references [1-3], we know that the existence and uniqueness of traveling wave solutions of reaction-diffusion equations in a finite interval can be completed by the following Schauder fixed point theorem.

Lemma 2.7 Let E be a closed convex set in a Banach space, let T : E → E be a continuous map such that TE is compact, then T has a fixed point.
It is easy to see that E is a closed convex set. In Banach space X, we denote the norm (f 1 , f 2 ) X = f 1 C(I l ) + f 2 C(I l ) . Since Uand Vare nonnegative, we have U ≥ 0, V ≥ 0 for any (U, V ) ∈ E. Lemma 2.8 For given (U 0 , V 0 ) ∈ E, there is a unique solution (U, V ) to the following boundary value problem: (
Proof The system is not a coupled system, thus we know that there are existence and uniqueness of U and V . By the definition of Uand V -, we know U -(-l) = V -(-l) = 0, U -(l) > 0, V -(l) > 0. Since the equation of V is a linear equation, it is easy to see the existence and uniqueness of V . Moreover, V + cV ≤ 0, V (±l) ≥ 0, by using the maximum principle, there is V > 0 on (-l, l). Next, we claim the existence and uniqueness of U. When V 0 is a given function, we see that the first equation of (2.19) is second order elliptic equation with boundary condition. Since the term U(1 -U -rV 0 1+U ) is Lipschitz continuous, according to the argument of regularity of the elliptic problem, the Sobolev imbedding theorem, and the contraction mapping principle, the existence of U is obtained. In addition, by applying the maximum principle, we can see that U > 0, U > 0 in (-l, l). Now, we define the mapping T :  From the third formula of (2.19), we get w 1 (l) = 0. In addition, from (2.5) and (2.20), there are w 1 (z) + cw 1 (z) ≤ 0 for all z ∈ (z 1 , l). According to the principle of maximum value, there is w 1 ≥ 0 in [z 1 , l]. And from the two conditions of w 1 ≥ 0 and (2.22) in [z 1 , l], we can get Similarly, we can conclude that there is V ≤ V + in I l .

Lemma 2.11 TE is compact.
Proof For a sequence {(U 0,n , V 0,n )} n∈N in E, let (U n , V n ) = T(U 0,n , V 0,n ). Because U + and Uare uniformly bounded on I l , and from Lemma 2.6, we know that the sequences {U n } and {V n } are also uniformly bounded on I l . Therefore, by applying the Arzela-Ascoli theorem, we obtain that {(U n , V n )} such that (U nj , V nj ) → (U, V ) uniformly on I l as i → ∞. Therefore, T(E) is compact in E. So T is precompact.
In view of Lemma 2.8, Lemma 2.9, Lemma 2.10, and Lemma 2.11, we prove that the mapping T satisfies all the assumptions of Lemma 2.7. Therefore, T has a fixed point. This fixed point is the nonnegative solution of system (2.18), so we can get the following result. (2.30)

The existence of traveling wave solution
In this section, we would like to show the main result of this paper. Proof Let {l n } n∈N be an increasing sequence in (z 1 , ∞) such that when n → ∞, there is l n → ∞, and let (U n , V n ) be the solution of system (2.19) and l = l n . And, for any given N ∈ N, because the function V + is bounded on [-l N , l N ], by ( Applying the Arzela-Ascoli theorem and the diagonal process, there is a subsequence When n → ∞, U and V in C 2 (R) are uniformly continuous in a compact set in R. It is easy to conclude that U > 0 on R. is convergence. Otherwise, it will deviate to ∞. And get U (∞) = ∞ from (3.3). Therefore, U(∞) = ∞, which contradicts the existence of U(∞), so U (∞) exists. At the same time, we can easily verify U (∞) = 0 from U(+∞) = 1. Similarly, V (∞) = 0 can be obtained by integrating the second expression of system (2.18) from 0 to z. Then, we would like to prove the existence of (U, V ) and 1 > U(-∞) ≥ 0, V (-∞) ≥ 0. Since U is increasing, and there is 0 ≤ U ≤ 1, thus U(-∞) exists and 0 ≤ U(-∞) ≤ 1, where U(-∞) = 1. If U(-∞) ≡ 1, according to the monotonicity of U, then U ≡ 1. By (2.18), we have V ≡ 0, which contradicts V ≥ V -> 0 in (z 1 , ∞). Therefore, U(-∞) = 1.
In order to prove the existence of V (-∞), we need to state that V ≤ 1 in R. By (1.1), we know bU + rV + c bU + rV + bU(1 -U) = 0.
Integrating the above equality from z to ∞, we obtain