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# Traveling wave solutions for a class of reaction-diffusion system

*Boundary Value Problems*
**volume 2021**, Article number: 33 (2021)

## Abstract

In this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

## Introduction

This paper deals with the existence and uniqueness of traveling wave solutions of the following reaction-diffusion system:

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 [5–8, 10–13] 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 \(\mathbb{R}\) to \(\mathbb{R}\) such that \(u(t, x):= U(x-ct), v(t, x):= V(x-ct)\) satisfies (2.1). Clearly, \(U(z),V(z)\) satisfy the following wave profile system:

There are boundary conditions \((U,V)(-\infty )=(0,1/r)\) and \((U,V)(+\infty )=(1,0)\). 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 \(\lambda +d\), there are

In addition, \(p(s)<0\), when \(s\in (\lambda,\lambda +d)\).

### Lemma 2.1

*The function*
\(V^{+}(z):=e^{-\lambda z}\)
*satisfies the equation*

for all \(z\in \mathbb{R}\).

### Proof

Since \(p(\lambda )=0\), we know

Let \(0<\gamma <\min \{\frac{c}{\delta },\lambda \}\), then \(c-\delta \gamma >0\), \(\gamma -\delta <0\). If \(z\rightarrow \infty \), then \(e^{(\gamma -\delta )z}\rightarrow 0\). There exists \(z_{0}>0\) such that

From the above formula we get

Set \(M=e^{\gamma z_{0}}\). Since \(\gamma,z_{0}>0\), we know \(M>1\). □

### Lemma 2.2

*The function* \(U^{-}(z):=\max \{0,1-Me^{-\gamma z}\}\) *satisfies the inequality*

*for all* \(z\neq z_{0}\).

### Proof

When \(z< z_{0}\), it can be concluded from the above inequality that \((-\infty,z_{0})\) is established on \(U^{-}\equiv 0\). When \(z>z_{0}\), there is \(U^{-}(z)=1-Me^{-\gamma z}\) and \(0< U^{-}<1\), so we have

By calculating (2.2), (2.4), and \(M>1\), we know

Thus, we can conclude that (2.3) holds. The proof is completed.

Let \(0<\eta <\min \{\gamma,d\}\). Since \(0<\eta <\gamma \), \(p(\lambda +\eta )<0\), select

Let \(z_{1}=\ln \frac{K}{\eta }\), \(z_{0}=\ln \frac{M}{\gamma }\), and \(K>M>1\), \(\eta <\gamma \), therefore \(z_{1}>z_{0}>0\). □

### Lemma 2.3

*The function* \(V^{-}(z):=\max \{0,V^{+}(z)-Ke^{-(\lambda +\eta )z}\}\) *satisfies the following inequality*:

*for all* \(z\neq z_{1}\).

### Proof

If \(z< z_{1}\), from (2.3), we can get that there is \(V^{-}\equiv 0\) on \((-\infty,z_{1})\). If \(z>z_{1}\), there are \(V^{-}=V^{+}-Ke^{-(\lambda +\eta )z}\) and \(U^{-}=1-Me^{-\gamma z}\), and by calculating it is easy to see that

Note

We know

Since \((V^{+})^{\prime \prime }+c(V^{+})^{\prime }+\frac{b}{2}V^{+}=0\), \(K>-Mb/(2p(\lambda +\eta ))\), \(1-Me^{-\gamma z}>0\), and \(\gamma >\eta \), 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* \(\phi _{1}\) *and* \(\phi _{2}\) *be the solution of second*-*order linear equation* \(L[y]:=y^{\prime \prime }-Ay^{\prime }+g(z)y=0\) *on* \([a,b]\) *such that*

*Then we have the following estimates of* \(\phi _{1}\) *and* \(\phi _{2}\):

*for all* \(z\in (a,b)\), *where* \(K_{1}= \Vert g \Vert _{C[a,b]}\).

### Proof

If \(g\le 0\), the Langsky matrix \(W(\phi _{1},\phi _{2})\) of \(\phi _{1}\) and \(\phi _{2}\) 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 $Y=\left(\begin{array}{c}y\\ {y}^{\prime}\end{array}\right)$, $B(z)=\left(\begin{array}{cc}0& 1\\ -g(z)& A\end{array}\right)$.

Considering \(z\in (a,b)\), we know that, for \(Y^{\prime }=B(z)Y\) in *Y*, the integral equation is satisfied

Thus

where \(\Vert \cdot \Vert \) represents the absolute norm, because of \(\Vert B(\cdot ) \Vert =\max \{ \vert g(\cdot ) \vert ,A+1\}\le K_{1}+A+1\), we can easily derive

Replacing *a* with *b* in (2.10), we get

Now, let $Y={\left(\begin{array}{cc}{\varphi}_{1}& {\varphi}_{1}^{\prime}\end{array}\right)}^{T}$, \(\Vert Y(z) \Vert = \vert \phi _{1}(z) \vert + \vert \phi _{1}^{\prime }(z) \vert \), \(\Vert Y(a) \Vert =1\), so (2.6) can be obtained from (2.11). Similarly, we let $Y={\left(\begin{array}{cc}{\varphi}_{2}& {\varphi}_{2}^{\prime}\end{array}\right)}^{T}$ in (2.12) to get (2.7). Note that we prove (2.8), applying the Abel formula and noting \(W(\phi _{1},\phi _{2})(b)=-\phi _{1}(b)\), we get

In order to estimate \(\phi _{1}(b)\), we make the second-order equation \(\rho (z)=\frac{1}{A}(e^{A(z-a)}-1)\) on \([a,b]\) have a unique solution \(\rho ^{\prime \prime }-A\rho ^{\prime }=0\). So, there is \(\rho (a)=0\), \(\rho ^{\prime }(a)=1\). When \(z\in (a,b)\), taking note of \(\rho \ge 0\), \(g\le 0\), we find that the function \(\psi:=\phi _{1}-\rho \) satisfies

According to the maximum principles, we get \(\psi \ge 0\), so \(\phi _{1}\ge \rho \) 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* \([a,b]\). *Let* \(w\in C([a,b])\cap C^{2}([a,b])\) *satisfy the differential equation*

*There is* \(w(a)=w(b)\). *If* \(-K_{1}\le g\le 0\), \(\vert h \vert \le K_{2}\) *on* \([a,b]\), *where* \(K_{1}\), \(K_{2}\) *are constant*. *There exist a positive constant* \(K_{1}\) *that depends only on the length of* \([a,b]\) *and a positive constant* \(K_{3}\) *such that* \(\Vert w \Vert _{C[a,b]}\le K_{2}K_{3}\) *holds*.

### Proof

First, we notice that both \(w^{\prime }(a+):=\lim_{z \to a+}w^{\prime }(z)\), \(w^{\prime }(b-):=\lim_{z \to b-}w^{\prime }(z)\) exist, such that \(\bar{z}\in (a,b)\) is fixed. Integrating the both sides of \(w^{\prime \prime }(z)+Aw^{\prime }(z)+g(z)w(z)=h(z)\) from *z̄* to *z*, we have

Since *w*, *g*, and *h* are right continuous at point *a*, thus \(w^{\prime }(a+)\) exists. Similarly, we also have the existence of \(w^{\prime }(b-)\). Let \(\phi _{1}\) and \(\phi _{2}\) be as shown in Lemma 2.4. For any \(z\in (a,b)\), integrating the inequality \(w^{\prime \prime }(z)\phi _{1}(z)+Aw^{\prime }(z)\phi _{1}(z)+g(z)w(z)\phi _{1}(z)=h(z) \phi _{1}(z)\) from *a* to *b*, we know

Similarly, we have

Since \(w(a)=w(b)=0\), multiplying (2.15) and (2.16) by \(\phi _{1}\), \(\phi _{2}\) respectively, we get

Thus

Since \(\vert h \vert \le K_{2}\), this implies

Finally, by virtue of (2.6), (2.7), (2.8), and the above inequality, we can get that \(\Vert w \Vert _{C[a,b]}\le K_{2}K_{3}\) holds. □

### Lemma 2.6

*Let* *A*, *g*, *and* *h* *be the same as in the previous lemma*. *Let* \(w\in C([a,b])\cap C^{2}([a,b])\) *satisfy* \(w^{\prime \prime }(z)+Aw^{\prime }(z)+g(z)w(z)=h(z)\) *in* \((a,b)\). *If* \(\Vert w \Vert _{C[a,b]}\le K_{0}\), *then there is a constant* \(K_{4}\), *only on* *A*, \(K_{0}\), \(K_{1}\), \(K_{2}\) *and interval length* \([a,b]\), *so there is*

### Proof

Let \(\phi _{1}\), \(\phi _{2}\) be the same as in Lemma 2.4. For any point \(z\in (a,b)\), multiplying (2.15), (2.16) by \(\phi _{1}^{\prime }\) and \(\phi _{2}^{\prime }\), respectively, we have

It can be concluded that

Under the assumptions of \(\vert h \vert \le K_{2}\) and \(\vert w \vert \le K_{0}\), 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

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* \(T:E\rightarrow E\) *be a continuous map such that* *TE* *is compact*, *then* *T* *has a fixed point*.

Let

It is easy to see that *E* is a closed convex set. In Banach space *X*, we denote the norm \(\Vert (f_{1},f_{2}) \Vert _{X}= \Vert f_{1} \Vert _{C(I_{l})}+ \Vert f_{2} \Vert _{C(I_{l})}\). Since \(U^{-}\) and \(V^{-}\) are nonnegative, we have \(U\ge 0\), \(V\ge 0\) for any \((U,V)\in E\).

### Lemma 2.8

*For given* \((U_{0},V_{0})\in E\), *there is a unique solution* \((U,V)\) *to the following boundary value problem*:

*In addition*, *the solution* \((U,V)\) *satisfies* \(U>0\), \(V>0\) *in* \((-l,l)\).

### 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 \(U^{-}\) and \(V^{-}\), 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, \(V^{\prime \prime }+cV^{\prime }\leq 0\), \(V(\pm l)\ge 0\), 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 \(U(1-U-\frac{r V_{0}}{1+U})\) 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 \(U>0\), \(U^{\prime }>0\) in \((-l,l)\).

Now, we define the mapping \(T:E\rightarrow X\) by \(T(U_{0},V_{0})=(U,V)\), \(\forall (U_{0},V_{0})\in E\), where \((U,V)\) is the unique solution to the boundary value problem (2.19). Obviously, any fixed point of *T* is the solution of problem (2.19). □

### Lemma 2.9

\(TE\subseteq E\).

### Proof

For given \((U_{0},V_{0})\in E\), denote

We claim to have \(V^{-}\le V\le V^{+}\) on \(I_{l}\). Due to \(0\le U^{-}\le U_{0}\le U^{+}\equiv 1\) and \(0\le V^{-}\le V_{0}\le V^{+}\), we have

Thus

and

for all *z* in \((-l,l)\). Let \(w_{1}=V-V^{-}\), note that \(V^{-}=0\) and \(V\ge 0\) in \([-l,z_{1}]\), therefore

From the third formula of (2.19), we get \(w_{1}(l)=0\). In addition, from (2.5) and (2.20), there are \(w_{1}^{\prime \prime }(z)+cw_{1}^{\prime }(z)\le 0\) for all \(z\in (z_{1},l)\). According to the principle of maximum value, there is \(w_{1}\ge 0\) in \([z_{1},l]\). And from the two conditions of \(w_{1}\ge 0\) and (2.22) in \([z_{1},l]\), we can get \(V^{-}\le V\) in \(I_{l}\). Similarly, we can conclude that there is \(V\le V^{+}\) in \(I_{l}\).

Next, we claim that \(U^{-}\le U\) in \(I_{l}\). Since \(U^{-}\equiv 0\) and \(U\ge 0\) on \([-l,z_{1}]\), thus

On the interval \((z_{0},l]\), we know

For simplicity, we denote \(\psi (\xi ):=\frac{\xi }{1+\xi }\) and \(w_{2}:=U-U^{-}\). According to (2.3) and (2.24), we have \(w_{2}^{\prime \prime }+cw_{2}^{\prime }+w_{2}(1-w_{2})-q(z)w_{2}\le 0\) on \((z_{0},l)\), where

By using of the mean value theorem, we know that *q* is nonnegative at \((z_{0},l)\). In view of (2.23) and (2.19), it is easy to see that \(w_{2}(z_{0})\ge 0\), \(w_{2}{l}=0\). Applying the principle of maximum value, we have \(w_{2}\ge 0\) on \([z_{0},l]\), thus \(U\ge U^{-}\) on \([z_{0},l]\).

Finally, we would like to prove \(U\le U^{+}\) on \(I_{l}\). Since \(U^{*}\equiv 1\) and \(V_{0}\ge 0\), we have

Similarly, we find \(U\le U^{+}\) on \(I_{l}\). □

### Lemma 2.10

*T* *is a continuous map*.

### Proof

For \((U_{0},V_{0})\) and \((\widetilde{U}_{0},\widetilde{V}_{0})\) in *E*, it implies

Let \(w_{1}:=U-\widetilde{U}\), it is easy to see that \(w_{1}^{\prime \prime }+cw_{1}^{\prime }+w_{1}(1-w_{1})+g(z)w_{1}=h_{1}(z)\) and \(w_{1}(-l)=w_{1}(l)=0\), where

and

*ψ* is the same as in Lemma 2.9. Since \(0\le U\), \(\widetilde{U}\le 1\), and \(0\le \xi \le 1\), we know \(0\le \psi ^{\prime }(\xi )\le 1\). By applying the mean value theorem, we find \(0\le \frac{\psi (U(z))-\psi (\widetilde{U}(z))}{U(z)-\widetilde{U}(z)} \le 1\), when \(U(z)\neq\widetilde{U}(z)\).

Note that \(\delta >0\), \(0\le V_{0}\le V^{+}\), \(0\le \psi ^{\prime }(U(z))\le 1\), we have \(-K_{1}\le g\le 0\), where \(K_{1}=r \Vert V^{+} \Vert _{C(I_{l})}\). In fact \(0\le \psi (\widetilde{U})\le 1\), it is easy to see \(\vert h_{1} \vert \le r \Vert V_{0}-\widetilde{V}_{0} \Vert _{C(I_{l})}\).

By applying Lemma 2.5, we know that there is a constant \(C_{1}\) that depends only on *δ*, *c*, \(K_{1}\), *l* such that

By the definition of \(w_{1}\), we know

Let \(w_{2}=V-\tilde{V}\) and \(\phi (\xi )=\frac{\xi }{1+\xi }\), we have \(w_{2}(-l)=w_{2}(l)=0\) and \(w_{2}^{\prime \prime }+cw_{2}^{\prime }=h_{2}(z)\), where \(h_{2}=-b(\phi (\widetilde{U}_{0})\widetilde{V}_{0}-\phi ( U_{0}) V_{0})\). This implies

Since \(0\le U_{0},\widetilde{U}_{0}\le 1\), by using the mean value theorem to

it implies

Using

and

by (2.27), we infer

Then, from Lemma 2.5, there is a constant \(C_{2}\) depending only on *c* and *l* such that

Together with the definition of \(w_{2}\), we get

By virtue of (2.25),(2.26), and (2.28), we have

where \(C_{3}=bC_{2} \Vert V^{+} \Vert _{C(I_{l})}+rC_{1}+bC_{2}\). For any given \(\epsilon >0\), we set \(0<\delta <\frac{\epsilon }{C_{3}}\). By (2.29), for \(\varepsilon >0\), there is \(\delta >0\) such that

if \(\Vert (U_{0},V_{0})- (\widetilde{U}_{0},\widetilde{V}_{0}) \Vert _{X}<\delta \) for any \((U_{0},V_{0}),(\widetilde{U}_{0},\widetilde{V}_{0})\in E\). Therefore, *T* is a continuous map. Thus, the proof is completed. □

### Lemma 2.11

*TE* *is compact*.

### Proof

For a sequence \(\{(U_{0,n},V_{0,n})\}_{n\in \mathbb{N}}\) in *E*, let \((U_{n},V_{n})=T{(U_{0,n},V_{0,n})}\). Because \(U^{+}\) and \(U^{-}\) are uniformly bounded on \(I_{l}\), and from Lemma 2.6, we know that the sequences \(\{U_{n}^{\prime }\}\) and \(\{V_{n}^{\prime }\}\) are also uniformly bounded on \(I_{l}\). Therefore, by applying the Arzela–Ascoli theorem, we obtain that \(\{(U_{n},V_{n})\}\) such that \((U_{nj},V_{nj})\rightarrow (U,V)\) uniformly on \(I_{l}\) as \(i\rightarrow \infty \). 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.

### Lemma 2.12

*System* (2.18) *has a solution* \((U,V)\) *on* \(I_{l}\), *and this solution satisfies*

## The existence of traveling wave solution

In this section, we would like to show the main result of this paper.

### Theorem 3.1

*System* (2.1) *has a unique traveling wave solution* \((U,V)\) *and*

### Proof

Let \(\{l_{n}\}_{n\in \mathbb{N}}\) be an increasing sequence in \((z_{1},\infty )\) such that when \(n\rightarrow \infty \), there is \(l_{n}\rightarrow \infty \), and let \((U_{n},V_{n})\) be the solution of system (2.19) and \(l=l_{n}\). And, for any given \(N\in \mathbb{N}\), because the function \(V^{+}\) is bounded on \([-l_{N},l_{N}]\), by (2.8), we find that the sequence \(\{{U_{n}}\}_{n\ge N}\), \(\{{V_{n}}\}_{n\ge N}\), \(\{ \frac{U_{n}V_{n}}{1+U_{n}}\}_{n\ge N}\) is uniformly bounded on \([-l,l]\). Then, by virtue of Lemma 2.6, the sequence \(\{U_{n}^{\prime }\}_{n\ge N}\), \(\{V_{n}^{\prime }\}_{n\ge N}\) is uniformly bounded on \([-l,l]\). From (2.19), we have the sequence \(\{U_{n}^{\prime \prime }\}_{n\ge N}\), \(\{V_{n}^{\prime \prime }\}_{n\ge N}\), \(\{U_{n}^{\prime \prime \prime } \}_{n\ge N}\), \(\{V_{n}^{\prime \prime \prime }\}_{n\ge N}\) is uniformly bounded on \([-l,l]\).

Applying the Arzela–Ascoli theorem and the diagonal process, there is a subsequence \(\{(U_{nj},V_{nj})\}\) of \(\{(U_{n},V_{n})\}\) such that

When \(n\rightarrow \infty \), *U* and *V* in \(C^{2}(\mathbb{R})\) are uniformly continuous in a compact set in \(\mathbb{R}\). It is easy to conclude that \(U^{\prime }>0\) on \(\mathbb{R}\). \((U,V)\) is the nonnegative solution of problem (2.1). From the definition of \(U^{-}\) and \(V^{+}\), if \(z\rightarrow \infty \), then \(U^{-}\rightarrow 1\), \(V^{+}\rightarrow 0\) and

The following proves that \((U,V)(-\infty )=(0,1/r)\) can be divided into the following steps.

First, we prove

Integrating the both sides of (2.18) from 0 to *z*, we have

Since \(U(+\infty )\) exists, we find that \(U^{\prime }(\infty )\) exists if and only if the integral

is convergence. Otherwise, it will deviate to ∞. And get \(U^{\prime }(\infty )=\infty \) from (3.3). Therefore, \(U(\infty )=\infty \), which contradicts the existence of \(U(\infty )\), so \(U^{\prime }(\infty )\) exists. At the same time, we can easily verify \(U^{\prime }(\infty )=0\) from \(U(+\infty )=1\). Similarly, \(V^{\prime }(\infty )=0\) 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 \(1>U(-\infty )\ge 0\), \(V(-\infty )\ge 0\). Since *U* is increasing, and there is \(0\le U\le 1\), thus \(U(-\infty )\) exists and \(0\le U(-\infty )\le 1\), where \(U(-\infty )\neq1\). If \(U(-\infty )\equiv 1\), according to the monotonicity of *U*, then \(U\equiv 1\). By (2.18), we have \(V\equiv 0\), which contradicts \(V\ge V^{-}>0\) in \((z_{1},\infty )\). Therefore, \(U(-\infty )\neq1\).

In order to prove the existence of \(V(-\infty )\), we need to state that \(V\le 1\) in \(\mathbb{R}\). By (1.1), we know

Integrating the above equality from *z* to ∞, we obtain

This implies that

Since \(U(\infty )=1,V(\infty )=0\), \(U'(\infty )=V'(\infty )=0\), we know that

Since

is convergence (by (3.11) and (3.12)), take a constant *K* such that

Let \(W=bU+rV-b-K\), since \(U\le 1\) and (2.30), we obtain

Note that \(\int ^{\infty }_{z}b U(1-U)\,d\tau \geq 0\), by (3.5), we get

Multiplying the above inequality by the term \(e^{cz}\), it is easy to see that \([e^{cz}W(z)]'\leq 0\), which implies that the function \([e^{cz}W(z)]\) is nonincreasing. Thus, if \(-\infty < z_{1}< z<+\infty \), then

Note that \(c>\lambda \), if \(z_{1}\rightarrow -\infty \), then \(W(z)\leq 0\). It is easy to see that \(bU+rV-b-K\leq 0\). Thus, we get that \(V\leq K/r\) in \(\mathbb{R}\).

Next, we prove the existence of \(V(-\infty )\) and \(V(-\infty )\ge 0\).

Since \(V(\infty )=0\) and \(V(z_{1}+1)\ge V^{-}(z_{1}+1)\), we use the mean value theorem to infer the existence of \(\xi _{1}\ge z_{1}+1\) such that \(V^{\prime }(\xi _{1})\ge 0\). Multiplying the second equation of (3.2) by the term \(e^{cz}\), we can easily obtain \([e^{cz}V^{\prime }(z)]^{\prime }=-e^{cz}\frac{bUV}{1+U}\leq 0\). Thus \(e^{cz}V^{\prime }(z)\) is nonincreasing. Since \(z>\xi _{1}\), \(e^{cz}V^{\prime }(z)\ge e^{c\xi _{1}}V^{\prime }(\xi _{1})>0\), then there is \(V^{\prime }<0\) on \([\xi _{1},\infty )\). Let \(\xi _{2}:=\inf \{z|V^{\prime }>0, \text{ in }[z,\infty )\}\). Set \(\xi _{2}\) be a finite number or \(\xi _{2}=-\infty \). If \(\xi _{2}=-\infty \), then there is \(V^{\prime }>0\) on \(\mathbb{R}\). Note that \(0\le V\le 1\), we know \(V(-\infty )\) exists and \(V(-\infty )>0\). If \(\xi _{2}\) is a finite number, then there is \(V^{\prime }(\xi _{2})=0\), which together with the monotonicity of \(e^{cz}V^{\prime }(z)\) leads to \(e^{cz}V^{\prime }(z)\leq e^{c\xi _{2}}V^{\prime }(\xi _{2})=0\), where \(z\ge \xi _{2}\). Therefore, there is \(V^{\prime }\leq 0\) on \((-\infty,\xi _{2}]\). We obtain that \(V(-\infty )\) exists and \(V(-\infty )\ge 0\). Next, we would like to prove

Under the condition \(U(\infty )=1\), \(U^{\prime }(\infty )=0\), integrating the first equation of (2.18) from *z* to ∞, we find

By \(U\ge 0\), \(U^{\prime }\ge 0\), we have

This implies that the improper integral

converges. Let \(z\rightarrow -\infty \), from (3.10) get \(U(-\infty )\) exist, we infer that \(U^{\prime }(-\infty )\) exists. In addition, because of \(U^{\prime }\ge 0\), then \(U^{\prime }(-\infty )\ge 0\) is derived. In fact, \(U^{\prime }(-\infty )=0\), if \(U^{\prime }(-\infty )>0\), then \(U(-\infty )=-\infty \), which contradicts the existence of \(U(-\infty )\).

Through a similar proof, we can also get \(V^{\prime }(-\infty )=0\) and

Next, we prove that \((U,V)(-\infty )=(0,1/r)\). Since \(U(-\infty ),V(-\infty )\) exists, by the improper integrals (3.12), we have

Similarly, by virtue of (3.11), we obtain

Since \(U(-\infty )\neq1\), from (3.13) and (3.14), we obtain that \((U,V)(-\infty )=(0,1/r)\). Therefore, the proof is completed. □

## Conclusion

In this paper, we discuss that system (1.1) has a unique translation traveling wave solution by the supersolution and subsolution method and the Schauder fixed point theorem. Moreover, the uniqueness wave solution \((U,V)\) of (1.1) satisfies \(U(\infty )=1, U(-\infty )=0, V(\infty )=0, V(-\infty )=1/r\).

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This paper is mainly completed by BW and ZY dealt with traveling wave solutions for a class of reaction-diffusion equations. All authors read and approved the final manuscript.

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Wang, B., Zhang, Y. Traveling wave solutions for a class of reaction-diffusion system.
*Bound Value Probl* **2021, **33 (2021). https://doi.org/10.1186/s13661-021-01508-7

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### Keywords

- Reaction-diffusion equation
- Traveling wave solution
- Existence and uniqueness