Global stability of traveling fronts of a diffusion system with the Belousov-Zhabotinskii reaction

1.   Introduction

This paper aims to study the stability of the following diffusion equation with the Belousov-Zhabotinskii (BZ) reaction:

where $r, b$ are positive parameters and $u, v$ corresponds to the concentration of bromous acid and bromide ion, respectively. The BZ chemical reaction is a famous oscillation reaction discovered by Belousov. About ten years later, Zaikin and Zhabotinskii observed traveling wave phenomena in such a chemical activity ^[32], and then Field and his coworkers ^[4,5] established a model to describe chemical wave in the BZ reaction. Later, based on experimental and numerical results, Murray ^[21,22] nondimensionalized the model to be system (1.1).

Since the traveling wave solution of (1.1) is found to be an appropriate mathematical tool to describe the chemical wave observed in the BZ reaction, it has attracted a lot of attention, for example, see the recent works ^{[3,6,7,15,26,27,33]} and references therein for the study of traveling wave solutions of (1.1).

It is natural to ask whether the aforementioned traveling wave solutions of (1.1) are stable or not, since the stability of traveling waves is also very important (but more difficult) to reaction diffusion equations. There are several methods to prove the stability of traveling wave solutions, among which three methods are frequently-used; see ^[8,24,31] for the spectral analysis method, see ^{[17,18,19,20]} for the weighted energy method, and see ^[1,25,29] for the squeezing technique combined with the comparison principle.

For scalar equations $u_{t} = \Delta u+f(u), x\in\Bbb{R}, \ t > 0,$ the stability of traveling waves has been well studied; see ^{[8,11,13,23,30,31]} and the references therein. For the systems of reaction diffusion equations, Kapitula ^[10] considered a semi-linear parabolic system. Using semigroup theory, Kapitula proved that the wave fronts are stable in polynomially weighted $L^{\infty}$ spaces, and the convergence speed is given by the detail semigroup estimate. By detail spectral analysis, Sattinger ^[23] proved that the traveling wave fronts were stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Kan-on and Fang ^[9] obtained the asymptotic stability of monotone traveling waves for a competition-diffusion system by using spectral analysis. Kessler and Levine ^[12] investigated linear stability as well as nonlinear stability of the traveling wave solutions in a piecewise linear Oregonator model arising in the BZ reaction, but they did not estimate the convergence speed of the traveling waves. Lv and Wang ^[16] studied the asymptotic stability of a cooperative Lotka-Volterra system by the weighted energy method and obtained the time decay rates, which is also valid for the BZ system. However, their results depend mainly on the condition that the initial perturbations are in a weighted $H^{1}$ space. Recently, Du et al. ^[2] and Wang et al. ^[28] also considered the stability of the BZ system with delay, but their stability results are based on the weighted energy method or the weighted spaces.

With all the above in mind, the purpose of this article is to study the asymptotic stability of traveling fronts of system (1.1) by the squeezing technique combined with the comparison principle. Note that in system (1.1), $r$ is a key parameter that decides the characteristic of the BZ system. Precisely, the BZ system is mono-stable if $r\in (0, 1]$, while it is bistable if $r > 1$.

In the current paper, we always assume that $r > 1$ and $b > 0$, i.e., we study the bistable case. Let $u_{1}(x, t) = u(x, t), u_{2}(x, t) = 1-v(x, t)$, $\mathbf{u} = (u_{1}, u_{2})$ and $\mathbf{F}(\mathbf{u}) = (f_{1}(\mathbf{u}), f_{2}(\mathbf{u})) = (u_{1}(1-r-u_{1}+ru_{2}), bu_{1}(1-u_{2}))$, then system (1.1) can be rewritten as

We should emphasize that the 'bistable' case here is not standard. Actually, it is easy to see that $(0, u_{2})$ is a steady state of (1.2) for any $u_{2}\in \mathbb{R}$, which means that (1.2) is degenerate at the equilibrium (0, 0).

It is known from ^[7,26] that system (1.2) admits a unique (up to translation) positive traveling front ($\mathbf{U}(\xi)$, $c$), $\mathbf{U}(\xi) = (U_{1}(\xi), U_{2}(\xi))$, satisfying $U_{1}(\xi) < U_{2}(\xi)$ and

The main result of this paper is stated as follows.

Theorem 1.1. Assume $b > 0$ and $r > 1$. If the initial value $\mathbf{u}_{0}(x)\in[\mathbf{0}, \mathbf{1}]$ satisfies

where $\lambda_{2} = c$, then the solution $\mathbf{u}(x, t;\mathbf{u}_{0})$ of (1.2) with $\mathbf{u}(x, 0;\mathbf{u}_{0}) = \mathbf{u}_{0}(x)$ satisfies

where $0 < \alpha_{2} < \alpha_{1} < 1,$ and $C, k > 0, \xi_{0}\in\Bbb{R}$ are some constants.

Remark: This result implies that the traveling front $\mathbf{U}(x+ct)$ is asymptotically stable under initial perturbations that decay as $|x|\rightarrow \infty$, but can be possibly large in any finite intervals. The convergence rate is exponential.

To use the comparison argument, we modify system (1.2). Let $\tilde{\mathbf{F}}(\mathbf{u}) = \mathbf{F} (\mathbf{u})+\mathbf{G}(\mathbf{u})$, where $\mathbf{G}(\mathbf{u}) = (g_1(\mathbf{u}), g_2(\mathbf{u}))$ with

It is easy to check that

and

Thus, the comparison principle works for the following Cauchy problem:

That is, if we denote the solution of (1.6) by $\tilde{\mathbf{v}}(x, t;\mathbf{v}_{0})$, then $\tilde{\mathbf{v}}(x, t;\mathbf{v}^{1}_{0})\leq \tilde{\mathbf{v}}(x, t;\mathbf{v}^{2}_{0})$ if $\mathbf{v}^{1}_{0}(x)\leq\mathbf{v}^{2}_{0}(x)$. It is also easy to see that $[\mathbf{0}, \mathbf{1}]$ is the invariant interval for the solution of (1.6), namely, if $\mathbf{v}_{0}(x)\in \; [\mathbf{0}, \mathbf{1}]$, then $\tilde{\mathbf{v}}(x, t;\mathbf{v}_{0})\in [\mathbf{0}, \mathbf{1}]$. Thus, for $\mathbf{v}_{0}(x)\in [\mathbf{0}, \mathbf{1}]$, the solution $\tilde{\mathbf{v}}(x, t;\mathbf{v}_{0})$ of (1.6) is also the solution of (1.2) with the same initial data, i.e., $\tilde{\mathbf{v}}(x, z, t;\mathbf{v}_{0})\equiv \mathbf{v}(x, z, t;\mathbf{v}_{0})$, where $\mathbf{v}(x, z, t;\mathbf{v}_{0})$ denotes the solution of (1.2) with initial data $\mathbf{v}_{0}$.

The outline of this paper is as follows. In Section 2, we give some notations and known results. In Section 3, we construct some super and sub-solutions for later use. The final section is devoted to the proof of the main theorem.

2.   Preliminary

First we introduce some notations.

For vectors $\mathbf{x}, \mathbf{y}\in \mathbb{R}^{2}$, we define their order relationships. We use $\mathbf{x} < \mathbf{y}$ to mean $x_{i} < y_{i}, i = 1, 2$, and $\mathbf{x}\leq\mathbf{y}$ to mean $x_{i}\leq y_{i}, i = 1, 2$. The interval is $[\mathbf{x}_{1}, \mathbf{x}_{2}]: = \{\mathbf{x}\in \mathbb{R}^{2}: \mathbf{x}_{1}\leq\mathbf{x}\leq \mathbf{x}_{2}\}$. Particularly, we denote $\mathbf{0} = (0, 0)$ and $\mathbf{1} = (1, 1)$.

Now, we list the asymptotic behaviors of the wavefront profile $\mathbf{U}(\xi)$ at the space infinity; see [26, Lemmas 13 and 14].

where $\lambda_{2} = c$ and $\lambda_{1}$ is the positive root of the equation $x^{2}-cx+1-r = 0$, and

where $\tilde{\zeta_{1}} = \frac{1}{2}(c-\sqrt{c^{2}+4})$ and $\zeta_{1} = \frac{1}{2}(c-\sqrt{c^{2}+4b})$. In the above, $A > 0$, $\sigma > 0$, $\xi_{0}$, $\xi_{1}$, $d$, and $d_{1}$ are appropriate constants. Since $\lambda_{1} > \lambda_{2}$, it is easy to see that

We can find two positive constants $L_{1}, L_{2}$ such that

In the following of this section, we give some notations. Denote $f_{ij}(\textbf{u}) = \frac{\partial f_{i}(\textbf{u})}{\partial u_{j}}$, and let

We introduce a vector $\textbf{q}: = (q_{1}, q_{2})\in(\mathbf{0}, \mathbf{1})$ and denote its transpose by $\textbf{q}^{T}$. Notice the fact $r > 1$, then we can find an appropriate $\textbf{q}$ with $q_{1} > r q_{2}$ such that

Since $D\textbf{F}(\textbf{u})$ is continuous in $\mathbf{u}$, we can fix a small number $\varepsilon\in (0, 1)$ such that

for any $\mathbf{u}\in[(1-\varepsilon)\textbf{1}, (1+\varepsilon)\textbf{1}]$.

Throughout this paper, we always denote

and

Obviously, $\Pi_{i}(\alpha_{i}\lambda_{2}) < 0$ for $i = 1, 2$; see Theorem 1.1 for $\alpha_{i}$.

3.   Super-sub-solutions and some technical lemmas

In this section, we establish the super and sub-solutions as well as some technical lemmas, which will be used in the proof of the main result.

Define the operator $\tilde{\mathcal{L}}$ by

Lemma 3.1. Let $r > 1$ and $b > 0$. Denote $\beta_{0}: = \min\left\{\frac{\min_{i = 1, 2}\left\{-\frac{1}{4}\Pi_{i}(\alpha_{i}\lambda_{2}) \right\}}{N_{1}+1}, \frac{q_{1}-rq_{2}}{16q_{1}}, \frac{b}{8}\right\}$, then for each fixed $\beta\in(0, \beta_{0}]$, there exist a positive number $\rho = \rho(\beta)$ and a $\delta = \delta(\rho)$ such that the functions defined by

are a super-solution and a sub-solution to (1.6) on $t\in[0, \infty)$, respectively, where

Proof. First, we prove that $\mathbf{v}^{+}$ is a super-solution, i.e.,

Let $\eta^{+} = x+ct+\xi+\rho\delta(1-e^{-\beta t})$. A direct computation gives that

For any $t\geq 0$, we have

since $\rho\delta\beta e^{-\beta t}U'_{i}(\eta^{+}) > 0$. Then, we consider three cases.

Case 1. $\eta^{+} < -X$ for $X > 0$ large enough.

By the fact $\lim_{x\rightarrow -\infty}U_{2}'(x) = 0$, there exists a constant $X_{1} > 0$ such that $U_{2}(\eta^{+})\leq \frac{1}{2}$ for any $\eta^{+}\leq-X_{1}$, and following this we have

This implies that $\tilde{f}_{i}(\mathbf{v}^{+}(x, t)) = f_{i}(\mathbf{v}^{+}(x, t))$ for $\eta^{+}\leq-X_{1}$ and $i = 1, 2.$

Moreover, since $\lambda_{2} = \lim\limits_{x\rightarrow -\infty}\frac{U'_{2}(x)}{U_{2}(x)}$ and $\lambda_{2}^{2} = \lim\limits_{x\rightarrow -\infty}\frac{U''_{2}(x)}{U_{2}(x)},$ we have

and

as $x\rightarrow -\infty$. Thus, there exists $X_{2} > 0$ large enough such that

and

for any $x < -X_{2}$.

For the reaction term, we have

where $\eta_{\tau_{i}}: = \textbf{U}(\eta^{+})+\tau_{i}\delta e^{-\beta t}{\cal U}^{\mathit{\pmb{\alpha}}}(\eta^{+})$ with $\tau_{i}\in (0, 1), i = 1, 2.$ If $i = 1,$

as $\eta^{+}\rightarrow -\infty.$ If $i = 2,$ by the fact that $\lim_{x\rightarrow -\infty}U_{2}^{\alpha_{1}-\alpha_{2}}(x) = 0$, we have

as $\eta^{+}\rightarrow -\infty.$ It follows that there exists $X_{3} > 0$ large enough such that

and

for any $\eta^{+} < -X_{3}.$ Take $X = \max\{X_{1}, X_{2}, X_{3}\}$.

Then, for any $\eta^{+} < -X$, (3.1)–(3.5) yield that

provided that $\beta\leq \min_{i = 1, 2}\left\{-\frac{1}{4}\Pi_{i}(\alpha_{i}\lambda_{2})\right\}$.

Case 2. $\eta^{+} > X'$ for $X' > 0$ large enough.

In this case, it is not difficult to see that

Recall the definition of $\tilde{f}_{i}$. We know

where $\eta_{\tau_{i}}: = \textbf{U}(\eta^{+})+\tau_{i}\delta e^{-\beta t}{\cal U}^{\mathit{\pmb{\alpha}}}(\eta^{+})$ with $\tau_{i}\in (0, 1), i = 1, 2.$ Since $\lim_{x\rightarrow \infty}U_{2}(x) = 1$, there exists $X'_{1} > 0$ large enough such that

Moreover, there exists $X'_{2} > 0$ large enough such that

for any $\eta^{+} > X'_{2}$, provided that $\delta\leq \varepsilon$. If $i = 1$, it follows from (2.3) that for any $\eta^{+} > X'_{2}$, we have

If $i = 2$, it follows (2.3) and (3.6) that for any $\eta^{+} > \max\{X'_{1}, X'_{2}\}$, we have

Recalling the asymptotic behaviors of $U_{2}$, we can choose a positive $X'_{3}$ large enough such that for any $\eta^{+} > X'_{3},$ it holds

Take $X' = \max\{X'_{1}, X'_{2}, X'_{3}\}$. Then, for any $\eta^{+} > X'$, we have

provided that

Case 3. $-X\leq\eta^{+}\leq X'$.

Let $u_{*}: = \min_{i = 1, 2}\min_{-X\leq x\leq X'}U'_{i}(x)$. It is easy to see that

Then, we have

provided that $\rho > \frac{1+N_{2}+2M_{0}+b}{\beta u_{*}}$.

Combing the three cases above, we have proved that $\mathbf{v}^{+}(x, t)$ is a super-solution. In the following, we prove that $\mathbf{v}^{-}(x, t)$ is a sub-solution.

Let $\eta^{-} = x+ct+\xi-\rho\delta(1-e^{-\beta t})$. Similarly, we have

We also divide the whole interval into three parts.

Case 1. $\eta^{-} < -Y$ for some $Y > 0$ large enough.

Recalling the definition of $\tilde{f}_{i}$, we have $\tilde{f}_{2}(\mathbf{v}^{-}(x, t)) = f_{2}(\mathbf{v}^{-}(x, t))$ and

as $\eta^{-}\rightarrow -\infty,$ since

Thus, there exists $Y_{1} > 0$ large enough such that

Take $Y = \max\{Y_{1}, X\}$. It then follows from (3.1)–(3.3), (3.5), and (3.7) that

provided that $\delta < 1/\rho$ and $\beta+ \beta N_{1}\leq \min_{i = 1, 2}\left\{-\frac{1}{4}\Pi_{i}(\alpha_{i}\lambda_{2})\right\}.$

Case 2. $\eta^{-} > Y'$ for some $Y' > 0$ large enough.

Since $\lim_{x\rightarrow \infty}\frac{U'_{2}(x)}{U_{2}(x)} = 0, \lim_{x\rightarrow \infty}\frac{U''_{2}(x)}{U_{2}(x)} = 0$, there exists $Y'_{1} > 0$ large enough such that

By the definition of $\mathbf{v}^{-},$ there exists a positive $Y'_{1}$ large enough such that $v_{1}(\eta^{-}) > 0$ for all $\eta^{-} > Y'_{1}$. This implies that $\tilde{f}_{1}(\mathbf{v}^{-}) = f_{1}(\mathbf{v}^{-})$. Take $Y' = \max\{Y'_{1}, X'\}$, and we have

provided that $\beta\leq \frac{1}{4}\min\left\{\frac{q_{1}-rq_{2}}{q_{1}}, b\right\}.$

Case 3. $Y\leq \eta^{-}\leq Y'$.

For simplicity, we still define $u_{*}: = \min_{i = 1, 2}\min_{-Y\leq x\leq Y'}U'_{i}(x)$. It is easy to see that

Thus, we have

provided that $\rho\beta u_{*}\geq N_{1}+ N_{2}+2M_{0}+r.$ Finally, let

and $0 < \delta < \delta_{0}: = \min\left\{\frac{q_{1}-rq_{2}}{16bq_{1}}, \frac{1}{8}, \varepsilon, \frac{1}{\rho} \right\}$. The proof is completed. □

We introduce an auxiliary lemma below before moving on.

Lemma 3.2. For any pair of super-solution and sub-solution ${{\mathit{\pmb{\omega}}}}^{\pm}(x, t)\in[\mathbf{-1}, \mathbf{2}]$ with ${{\mathit{\pmb{\omega}}}}^{+}(x, 0)\geq{{\mathit{\pmb{\omega}}}}^{-}(x, 0)$, there holds ${{\mathit{\pmb{\omega}}}}^{+}(x, t)\geq{{\mathit{\pmb{\omega}}}}^{-}(x, t)$ for $t\geq 0$ and $x\in \mathbb{R}$. Furthermore, one has

for any $x\in \Bbb{R}$ with $|x-z|\leq J$, where $\theta(J, t) = \frac{1}{\sqrt{4\pi t}}\exp\left\{-m_{0}t-\frac{(J+1)^{2}}{4t}\right\}$ with $m_{0}$ as a positive constant.

Proof. The comparison principle for parabolic systems implies $\mathit{\pmb{\omega}}^{+}(x, t)\geq\mathit{\pmb{\omega}}^{-}(x, t)$ directly. Let $\mathit{\pmb{\omega}}: = \mathit{\pmb{\omega}}^{+}-\mathit{\pmb{\omega}}^{-}.$ Then, $\mathit{\pmb{\omega}}$ satisfies

where $\eta_{\tau_{i}} = \tau_{i}\mathit{\pmb{\omega}}^{+}+(1-\tau_{i})\mathit{\pmb{\omega}}^{-}$ with $\tau_{i}\in (0, 1)$. In view of $\tilde{f}_{ij}(\eta_{\tau_{i}})\geq 0$ for $i\neq j$, we have

where $m_{0}: = \sup_{\mathbf{u}\in[\mathbf{-1}, \mathbf{2}], i = 1, 2}|f_{ii}(\mathbf{u})|$. Thus, we obtain

and following this we get (3.9). □

Lemma 3.3. Assume $r > 1$ and $b > 0$. Let $\tilde{\mathbf{v}}(x, t)\in [\mathbf{0}, \mathbf{1}]$ be a solution of (1.6). If $\tilde{\mathbf{v}}(x, t)$ satisfies

for some $T\geq 0$, where $\xi\in\mathbb{R}, h > 0$ are some constants, then for any $t\geq T+1$, there exists a positive number $\epsilon^{*}$ such that

where $\hat{\xi}(t), \hat{\delta}(t),$ and $\hat{h}(t)$ satisfy

and

Here, $\delta$ and $\rho$ are defined as in Lemma 3.1.

Proof. Obviously, $\mathbf{v}(x, t) = \tilde{\mathbf{v}}(x, t+T)$ is also a solution to (1.6) with $\mathbf{v}(x, 0) = \tilde{\mathbf{v}}(x, T)$. It then follows from Lemma 3.1 and the comparison principle that

for all $x\in \mathbb{R}, \ t\geq 0.$

Let $\epsilon_{1} = \frac{1}{2}\min_{i = 1, 2}\{U'_{i}(\eta):|\eta-cT-\xi|\leq 2\} > 0$ and $\bar{h} = \min\{1, h\}.$ By the mean value theorem, we obtain

Then, at least one of the following is true:

We consider case (ⅰ) only, and the case (ⅱ) is similar. Since $\alpha_{i} < 1$, we have

Thus, there exists a $M_{3} > 0$ such that

Let $J_{1} = M_{3}+c+2, z_{0} = -cT-\xi$ and $J_{2} = J_{1}+c+3$. Now, we divide $\mathbb{R}$ into two intervals:

If $x\in \Omega_{1}$, letting $t = 1$ and $\bar{\theta} = \theta(J_{1}+|z_{0}|, 1)$, then by (3.9) we have

Moreover, we have

where $\eta_{1} = x-z_{0}+c-\rho\delta(1-e^{-\beta})+\theta_{i}\cdot2\rho\epsilon^{*}\bar{h}$ with $\ \theta_{i}\in(0, 1)$ and

Clearly, $|\eta_{1}|\leq |x-z_{0}|+c+\rho\delta(1-e^{-\beta})+2\epsilon^{*}\bar{h}\leq J_{2}$, and, thus,

On the other hand, it is obvious that

Thus, by (3.10) we have

Then, for $x\in \Omega_{2}$, we have

Combining the above two cases, we know that

holds for all $x\in \Bbb{R}$ and $i = 1, 2.$ Denote $\eta_{2} = 2\rho\epsilon^{*}\bar{h}-\rho\delta(1-e^{-\beta})$. Then, the comparison principle implies that

for all $t\geq T+1,$ where

and

A direct computation gives that

Similarly, for any $t\geq T+1$, we have

where

It is easy to verify that

The proof is completed. □

4.   Asymptotic stability of traveling fronts

We prove the main result in this section.

Lemma 4.1. Let $\mathbf{u}_{0}(x)\in C(\mathbb{R}, [\mathbf{0}, \mathbf{1}])$ satisfy (1.4). Then, for some constants $z_{0}, z_{1}, z_{2}$, the solution $\tilde{\mathbf{v}}(x, t;\mathbf{u}_{0})$ of (1.6) satisfies

for all $x\in \mathbb{R}, \ t > 0,$ where $q_{i}(x, t) = q_{0, i}e^{-\epsilon t}\min\{e^{\alpha_{i}\lambda_{2}(x+ct-z_{0})}, 1\}$ with $q_{0, i} = q_{0}q_{i}, i = 1, 2,$ and

Proof. Let $z = x+ct,$ and define

where $q_{i}^{\pm}(x, t) = q_{0, i}e^{-\epsilon t}\min\{e^{\alpha_{i}\lambda_{2}(z-z_{0}\pm\eta(0))}, 1\}, \epsilon > 0$ will be chosen later, and $\eta(t)$ is bounded and to be chosen so that $\eta'(t) > 0$. To prove (4.1), we need only to prove that $\mathit{\pmb{\omega}}^{\pm}(x, t) = (\omega_{1}^{\pm}(x, t), \omega_{2}^{\pm}(x, t))$ are a pair of super and sub-solutions by the comparison principle. Precisely, we need to prove that

We give the proof for $\mathit{\pmb{\omega}}^{+}(x, t)$ first. A direct calculation gives that

and

Take $\delta\in \left(0, \varepsilon\right]$ small enough, and consider three cases.

Case 1. $U_{i}(z+\eta(t))\in[\delta, 1-\delta], i = 1, 2$.

Define $C_{\delta}: = \min_{i = 1, 2}\min_{U_{i}(x)\in [\delta, 1-\delta]}U'_{i}(x)$. Then, we have

provided that $\eta'(t)C_{\delta}\geq \left(\epsilon+2M_{0}+\max\{b, r\}\right)e^{-\epsilon t}$.

Case 2. $U_{i}(z+\eta(t))\geq 1-\delta$.

By the monotonicity of $U_{i}(\cdot)$, we know $z+\eta(t)\geq U_{i}^{-1}(1-\delta)$. It follows that there exists a constant $z_{0}$ such that $z\geq z_{0}-\eta(0)$. Thus, we have

It is obvious that

Recalling (2.3), we have

provided that $\epsilon\leq \frac{1}{2} \min\left\{\frac{q_{1}-rq_{2}}{2q_{1}}, \frac{b}{2}\right\}$ and $q_{0}\leq \frac{1}{2b} \min\left\{\frac{q_{1}-rq_{2}}{2q_{1}}, \frac{b}{2}\right\}$.

Case 3. $U_{i}(z+\eta(t)) < \delta$.

Obviously, $z\leq z_{0}-\eta(0)$ in this case for the same $z_{0}$ defined in Case 2. Then, we have

Recalling (2.1), we have

as $z\rightarrow -\infty$. Thus, if $i = 1$, it follows that

Moreover,

Thus, if $i = 2$,

Combing the facts above and taking $\delta > 0$ small enough, we have

provided that $\epsilon\leq \min_{i = 1, 2}\left\{-\frac{1}{2}\Pi_{i}(\alpha_{i}\lambda)\right\}.$

Now we need to prove that $\mathbf{u}_{0}(x)\leq {\mathit{\pmb{\omega}}^{+}(x, 0)}$ for all $x\in \mathbb{R}$. In fact, it follows from (1.4) for any $q_{0, i} > 0$ that there exists a positive constant $M$ large enough such that for $x < -M$, it holds

Using (1.4) again, we can find a positive constant $M'$ large enough such that for $x > M'$, it holds

For the case $-M\leq X\leq M'$, we can choose $\eta(0)$ sufficiently large to guarantee $\omega^{+}_{i}(x, 0)\geq u_{0, i}(x)$. Therefore, $\mathit{\pmb{\omega}}^{+}(x, t)$ is a super-solution to (1.6).

In the following, we prove that $\mathit{\pmb{\omega}}^{-}(x, t)$ is a sub-solution. We have

and

Similar to the proof for $\mathit{\pmb{\omega}}^{+}(x, t)$, we consider three cases.

Case 1. $U_{i}(z-\eta(t))\in[\delta, 1-\delta], i = 1, 2$.

Then, we have

Case 2. $U_{i}(z-\eta(t)) > 1-\delta, i = 1, 2$.

Then, there exists a $z_{0}$ such that $z\geq z_{0}+\eta(0)$. It follows that $q_{i}^{-}(x, t) = q_{0, i}e^{-\epsilon t}$ and

for $\epsilon\leq \frac{1}{2} \min\left\{\frac{q_{1}-rq_{2}}{2q_{1}}, \frac{b}{2}\right\}$.

Case 3. $U_{i}(z-\eta(t)) < \delta, i = 1, 2$.

We have $z\leq z_{0}+\eta(0)$, $q_{i}^{-}(x, t) = q_{0, i}e^{-\epsilon t}e^{\alpha_{i}\lambda_{2}(z-z_{0}-\eta(0))}$ and

Then, a similar discussion as in Lemma 3.1 for $\mathbf{v}^{-}$ yields that

for $\epsilon\leq \min_{i = 1, 2}\left\{-\frac{1}{2}\Pi_{i}(\alpha_{i}\lambda_{2})\right\}$. To complete the proof, we need only to prove that $\mathbf{u}_{0}(x)\geq {\mathit{\pmb{\omega}}^{-}(x, 0)}$. In fact, by (1.4), there exists two positive numbers $\tilde{M}$ and $\tilde{M}'$ large enough such that

and

For $-\tilde{M}\leq x\leq\tilde{M}'$, we can choose $\eta(0)$ large enough such that $\omega_{i}^{-}(x, 0)\leq u_{0, i}(x)$. Thus, $\mathit{\pmb{\omega}}^{-}(x, t)$ is a sub-solution. The proof is completed. □

Lemma 4.2. Assume the assumptions of Lemma 4.1 hold. Then, for any $\delta\in(0, 1)$, there exist $T = T(\delta)\geq 0$, $\xi$, and $h\in\Bbb{R}$ such that

for $x\in\Bbb{R}$ and $t\geq T$, $i = 1, 2$.

Proof. Recall the asymptotic behaviors of $\mathbf{U}$. We know $U_{2} < 1$ and $U_{2}^{\alpha_{i}}(x)\sim A^{\alpha_{i}}e^{\alpha_{i}\lambda_{2}x}$ as $x\rightarrow -\infty$. In view of Lemma 4.1, there exists a constant $M > 0$ such that

for $x\in \Bbb{R}, t > 0$. For any $\delta\in (0, 1),$ we can choose an appropriate $T\geq0$ such that $Me^{-\epsilon T}\leq \delta$. Moreover, let $\xi = -z_{1}$ and $h = z_{1}-z_{2}$. Then, (4.2) follows. □

Now, we prove the main result.

Proof of Theorem 1.1. Let $\beta, \rho$, and $\delta$ be as in Lemma 3.1. Let $\epsilon^{*}$ be defined in Lemma 3.3. Further, we choose a $\delta^{*} = \frac{1}{4}\epsilon^{*} < 1$, then it follows that $0 < k^{*}: = \rho(\epsilon^{*}-2\delta^{*}) < \frac{1}{4}$. Fix a $t^{*}\geq 1$ such that

First, we prove two claims.

Claim Ⅰ. There exist $T^{*} > 0$ and $\xi^{*}$ such that

for $x\in \Bbb{R}.$

Indeed, by Lemma 4.2, there exist $T = T(\delta^{*})\geq0$, $\xi$, and $h$ such that

for $x\in \Bbb{R}.$ If $h\leq 1$, then by the monotonicity of $\mathbf{U}(\cdot)$, Claim Ⅰ holds. If $h > 1$, we can choose an integer $N\geq 1$ such that $0 < h-Nk^{*}\leq k^{*} < 1.$ Then, (4.4) and Lemma 3.3 together with the choice of $k^{*}$ and $t^{*}$ imply that

where

and

Applying Lemma 3.3 again, we conclude that (4.5), with $t^{*}+T$ replaced by $Nt^{*}+T$, holds for some $\hat{\xi}\in\Bbb{R}, 0 < \hat{\delta}\leq \delta^{*}(1-k^{*})^{N}$ and $0\leq \hat{h}\leq h-Nk^{*}\leq 1.$ Let $T^{*}\! = \!Nt^{*}+T, \xi^{*}\! = \!\hat{\xi}$. By the monotonicity of $\mathbf{U}(\cdot)$, (4.3) holds.

Claim Ⅱ. Define $p\! = \!\rho(2\epsilon^{*}+\delta^{*}), T_{m}\! = \!T^{*}+mt^{*}, \delta^{*}_{m}\! = \!(1-k^{*})^{m}\delta^{*}$, and $h_{m}\! = \!(1-k^{*})^{m}, m\geq 0.$ Then, there exists a sequence $\{\hat{\xi}_{m}\}_{m = 0}^{\infty}$ with $\hat{\xi}_{0} = \xi^{*}$ such that

and

We prove Claim Ⅱ by mathematical induction. Clearly, Claim Ⅰ implies that (4.6) holds for $m = 0$. Suppose that (4.6) holds for $m = l\geq0$. Now, we are going to prove that (4.6) holds for $m = l+1$. Let $T = T_{l}$, $\xi = \hat{\xi}_{l},$ $h = h_{l}$, $\delta = \delta^{*}_{l}$ and $t = T_{l}+t^{*} = T_{l+1}\geq T_{l}+1$, then Lemma 3.3 yields that

where

and

Choose $\hat{\xi}_{l+1} = \hat{\xi}$. Then, we have

Thus, (4.6) holds for all $m\geq0.$

Now, we are ready to prove the main result. For $t\geq T^{*}$, let $m = \left[\frac{t-T^{*}}{t^{*}}\right]$ be the largest integer not greater than $\frac{t-T^{*}}{t^{*}}$, and define

and

Then, $T_{m} = T^{*}+mt^{*} < t < T^{*}+(m+1)t^{*} = T_{m+1}$. In view of (4.6), one has

for all $t\geq T^{*}$ and $x\in\Bbb{R}.$ Set $k: = -\frac{1}{t^{*}}\ln(1-k^{*})$ and $q = : e^{-\frac{t^{*}+T^{*}}{t^{*}}\ln(1-k^{*})}$. Since $0\leq m\leq \frac{t-T^{*}}{t^{*}} < m+1,$ we have

Thus,

and

It follows that for all $t'\geq t\geq T^{*}$, it holds

where $n = \left[\frac{t'-T^{*}}{t^{*}}\right]$. Obviously, $n\geq m.$ It follows from (4.10) that $\xi_{0}: = \lim_{t\rightarrow \infty}\xi(t)$ is well-defined. Letting $t'\rightarrow \infty$, we obtain

Since $\xi(t)$ and $h(t)$ are bounded, the asymptotic properties of $\mathbf{U}$ guarantee that

and

are well-defined and finite. Let $C_{0} = \max\{Q_{1}, Q_{2}\}$. Then, (4.7) gives that

Thus, (1.5) follows from (4.8)–(4.12). The proof is complete.

5.   Conclusions

In this paper, we investigated the asymptotic stability of traveling fronts of a diffusion system with the BZ reaction. By the squeezing technique and the comparison principle, we proved that if the initial perturbation decays to zero at the space infinity, then the perturbed solution converges to the traveling front of the system (1.1) with an exponential rate as $t\rightarrow +\infty$. Due to the degeneracy of (1.1) at the equilibrium (0, 0), the initial perturbation as $x\rightarrow -\infty$ was actually relaxed. It was only asked to decay to zero under an exponential weight. In fact, the degeneracy is caused by the second equation of (1.1). That is why we can derive the stability result in the form of (1.5), in which $U_{2}$ was taken to be the denominator at the left side of the inequality. In this point of view, we are able to present a little bit of contributions to the stability analysis of the BZ system.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No: 11901330).

Conflict of interest

The author declares no conflict of interest.