Multidimensional stability of V-shaped traveling fronts in bistable reaction-diffusion equations with nonlinear convection

1.   Introduction

In this paper, we study the multidimensional asymptotic stability of two-dimensional V-shaped traveling fronts of the following reaction-diffusion equation with nonlinear convection term:

where $\Delta = \partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_{n-2}^2+\partial^2/\partial y^2+\partial^2/\partial z^2$ and $n\geq3$. We assume that the initial value $u_0\in BUC^1(\mathbb{R}^n)$. In (1.1), $\left(g(u)\right)_z$ is a nonlinear convection term and the function $f\in C^2(\mathbb R)$ is the reaction term. Suppose $f$ satisfies the following assumption:

${\rm(F)}$ (ⅰ) $f(0) = f(1) = 0$, $f'(0) < 0, \ f'(1) < 0$;

(ⅱ) $\{r\in[0, 1]:f(r) = 0\} = \{0, \lambda, 1\}$ with $f'(\lambda) > 0$;

(ⅲ) $\int_0^1f(r){\rm d}r > 0$;

(ⅳ) $f(r) < 0, \ f'(r) < 0$ for $r > 1$; $f(r) > 0, \ f'(r) < 0$ for $r < 0$.

The assumption (F) implies that the reaction term $f$ is of bistable type. A typical example of such $f$ is the cubic function $f(u) = u(u-a)(1-u)$, where $a\in(0, \frac{1}{2})$ is a given number. We also assume the flux $g$ satisfies the following assumption:

${\rm(G)}$ $g(r)\in C^{2+\gamma_0}(\mathbb{R})$, $\gamma_0\in (0, 1)$; $g''(r)\leq0$ for $r\in[0, 1]$.

Examples of such $g$ are $g(u) = -\rho u^2$ and $g(u) = \rho u(1-u)$, where $\rho > 0$ is a constant. From the assumption (G), we deduce that there exist positive constants $l_1$ and $l_2$ such that

For each $\theta\in\left[0, \pi\right)$, it follows from Crooks and Toland ^[1] that there exist a function $U_\theta(\cdot)\in C^2\left(\mathbb{R}\right)$ and a constant $c_\theta$ satisfying

Let $\vec{e}_\pm = (0, \cdots, 0, \pm\cos\theta, \sin\theta)\in S^{n-1}$. Then the functions

and

are planar traveling fronts of (1.1) with wave speed $c_\theta$ along the directions $\vec{e}_+$ and $\vec{e}_-$, respectively. In particular, the profile function $U_\theta(\cdot)$ is unique up to a translation and the wave speed $c_\theta$ is unique. It is clear that $c_0 > 0$ due to the assumption (F). However, due to the convection term, it is not necessary that $c_\theta > 0$ for $\theta\in (0, \pi)$. For more results on traveling wave front of (1.1) with various nonlinearity $f$, we refer to ^{[2,3,4,5,6,7,8,9]}.

Recently, we studied nonplanar traveling fronts of (1.1) with $n = 2$. Assume that (F) and (G) hold. Fix $\theta\in(0, \frac{\pi}{2})$ satisfying the following assumption:

${\rm(C)}$ $c_\theta+g'(r)\sin\theta > 0$ for any $r\in[0, 1]$.

In our recent paper ^[10], we proved the existence of V-shaped traveling fronts to (1.1) in $\mathbb{R}^2$, namely, there exists a function $V(\cdot, \cdot)\in C^2\left(\mathbb{R}^2\right)$ satisfying

where $s_\theta = \frac{c_\theta}{\sin\theta}$. In addition, for any initial value $u_0(y, z)\in BUC^1(\mathbb{R}^2)$ with $u_0\geq U_\theta\left(|y|\cos\theta+z\sin\theta\right)$ and

the solution $u(y, z, t; u_0)$ of (1.1) satisfies

Here we would like to point out that the last results have been obtained when the convection term is absent, see ^[11]. In fact, nonplanar traveling fronts of (1.1) without convection have been extensively studied in recent years, see ^{[12,13,14,15,16,17,18,19,20,21,22,23,24,25]}. For reaction-diffusion system and time periodic reaction-diffusion equation, we refer to ^{[26,27,28,29,30,31,32,33,34]}. For more results on non-planar traveling wave solutions of reaction-diffusion equations, we refer to ^{[35,36,37,38]}.

It is clear that the asymptotic stability of the V-shaped traveling front $V(y, z)\in C(\mathbb{R}^2)$ of (1.1) was established in $\mathbb{R}^2$, see ^[10]. In this paper, we further consider the multidimensional stability of the V-shaped traveling front $V(y, z)$, namely, we consider the stability of $V(y, z)$ in $\mathbb{R}^n$ with $n\geq3$. It should be pointed out that the multidimensional stability of planar traveling fronts of (1.1) without convection has been studied by many authors, see ^{[39,40,41,42,43]} for the Allen-Cahn equation. In ^[39,40,43], it is assumed that the initial perturbations are sufficiently small and decay to zero at space infinity. In ^[42] the authors proved the asymptotic stability under any (possibly large) initial perturbations that decay to zero at space infinity. Furthermore, they proved the asymptotic stability of planar traveling fronts for almost periodic perturbation. Moreover, the existence of a solution that oscillates permanently between two planar waves was shown, which implies that planar traveling fronts are not asymptotically stable under general perturbations. Motano and Nara ^[41] studied how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. They showed that the planar front is asymptotically stable in $L^{\infty}(\mathbb{R}^n)$ under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. Roquejoffre and Roussier-Michon ^[44] considered the large time behavior of planar traveling fronts and showed that the dynamics of planar fronts are similar to that of the heat equation. More recently, Sheng et al. ^[28] and Cheng and Yuan ^[45] studied the multidimensional stability of V-shaped traveling fronts and pyramidal traveling fronts of the Allen-Cahn equation by using the method of ^[42], respectively.

Following (1.4) and (1.5), we know that the asymptotic stability of V-shaped traveling fronts $V(y, z)$ of (1.1) in ^[10] was established in $\mathbb{R}^2$ under the initial value $u_0(y, z)$ satisfying that $u_0(y, z)-V(y, z)$ decays to zero as $|y|+|z|\to \infty$. In this paper we use the method of ^[42] to study the stability of V-shaped traveling fronts $V(y, z)$ of (1.1) under the initial value $u_0(x, y, z)$ in $\mathbb{R}^n$ with $n\geq3$, where $x\in\mathbb{R}^{n-2}$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$. In contrast to that in ^[10], in this paper we deal with the following two cases:

Case A: The initial perturbation $u_0(x, y, z)-V(y, z)$ decays to zero as $|x|+|y|+|z|\rightarrow\infty$.

Case B: The initial perturbation $u_0(x, y, z)-V(y, z)$ does not necessarily decay to zero as $|x|+|y|+|z|\rightarrow\infty$.

In the remainder of this paper we always assume that $\rm(F)$ and $\rm(G)$ hold and $\theta\in \left(0, \frac{\pi}{2}\right)$ satisfies the assumption $\rm(C)$. Let $(U_\theta(\cdot), c_\theta)$ be defined by (1.3) and let $s_\theta = \frac{c_\theta}{\sin\theta}$. Let $V(y, z)$ be defined by (1.4). For the sake of convenience, in the sequel we always denote $(U_\theta(\cdot), c_\theta)$ and $s_\theta$ by $(U (\cdot), c )$ and $s$ respectively. In the following we first consider the asymptotic stability of $V(y, z)$ in Case A.

Theorem 1.1. Let $n\geq 3$. Assume that the initial value $u_0(x, y, z)\in BUC^1(\mathbb{R}^n)$ and satisfies

Then the solution $u(x, y, z, t)$ of (1.1) satisfies

Theorem 1.1 implies that the V-shaped traveling front $V(y, z+st)$ is asymptotically stable under any initial perturbations that decay to zero as $|x|+|y|+|z|\rightarrow\infty$. The following theorem gives the convergence rate for (1.7) when the initial perturbation belongs to $L^1$ in a certain sense.

Theorem 1.2. Let $n\geq 3$. Assume that the initial value $u_0(x, y, z)$ of (1.1) is given by

for some function $v_0\in L^1(\mathbb{R}^{n-2})\cap L^\infty(\mathbb{R}^{n-2})\cap BUC^1(\mathbb{R}^{n-2})$. Then the solution $u(x, y, z, t)$ of (1.1) satisfies

where $C > 0$ is a constant depending on $f, \ g, \|v_0\|_{L^1(\mathbb{R}^{n-2})}$ and $\|v_0\|_{L^\infty(\mathbb{R}^{n-2})}$.

The following proposition shows that the convergence rate (1.9) is optimal in some sense.

Proposition 1.3. Let $n\geq 3$. Let $u_0$ be defined as in (1.8) and assume that $v_0$ either satisfies $v_0\geq0, \ v_0\not\equiv0$ or $v_0\leq0, \ v_0\not\equiv0$. Then there exist constants $C_1 > 0$ and $C_2 > 0$ such that

Remark 1.4. Theorem 1.1 and 1.2 show that the V-shaped traveling front is not only asymptotically stable, but also algebraically stable under certain perturbations. Furthermore, Proposition 1.3 also implies that this convergence rate is optimal in some sense, that is, the convergence rate is not faster than $O\left(t^{-\frac{n-2}{2}}\right)$.

Next, we state our results in Case B. Firstly, we show that the V-shaped traveling front is also asymptotically stable if the initial value $u_0(x, y, z)$ satisfies some certain assumptions in Case B.

Theorem 1.5. Let $n\geq 3$. Suppose that the initial value $u_0(x, y, z)\in BUC^1(\mathbb{R}^n)$ of (1.1) satisfies

where $\hat{u}_0\left(y, z\right)\in BUC^1(\mathbb{R}^n)$ satisfies

Then the solution $u(x, y, z, t)$ ofx (1.1) satisfies

Secondly, we present a result on the existence of a solution of (1.1) that oscillates permanently between two V-shaped traveling fronts.

Theorem 1.6. Let $n = 3$. Then for any $\delta > 0$, there exists a bounded function $v_0^*(x)\in BUC^1(\mathbb R)$ with $\|v_0^*\|_{L^\infty(\mathbb R)} = \delta$ such that the solution $u(x, y, z, t)$ of (1.1) with $u(x, y, z, 0) = V\left(y, z+v_0^*(x)\right)$ satisfies

where $t_m = m(m!)^2/4$.

Remark 1.7. A typical example of initial functions satisfying the requirement of Theorem 1.5 is that $u_0(x, y, z) = V\left(y, z+\psi(z)v(x)\right)$ in $(x, y, z)\in\mathbb{R}^3$, where the functions $v(x)$ and $\psi(z)$ respectively satisfy the following conditions:

(ⅰ) $v(x)\in BUC^1(\mathbb R^{n-2})$ is a periodic function in $\mathbb{R}^{n-2}$ and $v(x) > 0$;

(ⅱ) $\psi(z)\in C^1(\mathbb R)$, $\psi(z) > 0$ and $\lim\limits_{|z|\rightarrow\infty}\psi(z) = 0$.

Since $v(x) > 0$ is a periodic function, there exists a positive constant $v_{\max}$ such that $v_{\max} = \max_{x\in\mathbb R^{n-2}}v(x)$. Take $\hat{u}_0(y, z) = V\left(y, z+\psi(z)v_{\max}\right)$. Then $V(y, z)\leq u_0(x, y, z)\leq\hat{u}_0(y, z)$ and (1.11) holds. In addition, it is clear that the initial function $u_0(x, y, z)$ does not decay when $|x|+|y|+|z|\rightarrow\infty$. Theorem 1.5 shows that even if in Case B, the V-shaped traveling front is still asymptotically stable under some further assumptions on the initial value. On the other hand, Theorem 1.6 implies that even very small perturbations to the V-shaped traveling front $V(y, z+st)$ can give rise to permanent oscillation, hence the V-shaped traveling front is not asymptotically stable under general bounded perturbations. It can be viewed as a counter-example to the asymptotic stability of V-shaped traveling fronts in Case B. From the viewpoint of dynamical system, Theorem 1.6 gives two $\omega$-limit points of the solution $u(x, y, z, t)$ in the $L_{\text{loc}}^\infty(\mathbb{R}^n)$-topology. From the above discussion, we conclude that the V-shaped traveling fronts may be stable or unstable in Case B.

The rest of the paper is organized as follows. In Section 2, we study Case A, namely, we prove Theorem 1.1 and 1.2. In the proofs, we use the moving coordinated with speed $s$ so that the V-shaped traveling fronts can be viewed as stationary states. Setting

the Eq (1.1) can be rewritten as

where $\Delta = \partial^2/\partial x_1^2+\cdots+\partial^2/\partial x^2_{n-2}+\partial^2/\partial y^2+\partial^2/\partial \chi^2$. For convenience, we denote $w(x, y, \chi, t)$ as $u(x, y, z, t)$ and consider the problem of the form

The global existence of a unique solution $u(x, z, t; u_0)$ of the Eq (1.13) follows from [46,Theorem 7.1.2,Propositions 7.1.9 and 7.1.10 and Remark 7.1.12] and the assumptions (F) and (G), see also [2,Proposition A.3 and Theorem A.7]. In particular, $u(t; u_0)(\cdot) \in C^1\left((0, \infty), BUC(\mathbb{R}^2)\right)\cap C \left((0, \infty), BUC^2(\mathbb{R}^2)\right) \cap C \left([0, \infty), BUC^1(\mathbb{R}^2)\right)$, where $u(t; u_0)(x, z): = u(x, z, t; u_0)$. It is clear that, for each $\xi\in\mathbb R$, the function $V(y, z+\xi)$ is a stationary solution for problem (1.13). In this section, we also show that the convergence rate in Theorem 1.1 and Theorem 1.2 is optimal in some sense, namely, we prove Proposition 1.3. In Section 3, we consider Case B. Firstly, we prove that the V-shaped traveling fronts are asymptotically stable under certain assumptions of the initial value, namely, we prove Theorems 1.5. Secondly, using the supersolutions and subsolutions constructed in Section 2, we prove that the V-shaped traveling fronts are not asymptotically stable, namely, we prove Theorem 1.6.

2.   Stability of V-shaped fronts in Case A

In this section, we consider Case A and prove the asymptotic stability of V-shaped traveling fronts under perturbations that decay to zero as $|x|+|y|+|z|\rightarrow\infty$. We first state some known results of the curvature flow problem in ^[47], see also ^[42].

The mean curvature flow for a graphical surface $\Psi(x, t)$ on $\mathbb{R}^{n-2}$ is given by the Cauchy problem of the form

If the first and second derivatives of $\Psi$ with respect to $x$ are all bounded on $\mathbb{R}^{n-2}$, then we take some large constant $k > 0$ such that

Clearly, $\Psi(x, t)$ is a subsolution of the following Cauchy problem:

Taking $w(x, t) = e^{kv^+(x, t)}$, we have

Then, from the solution of a standard heat equation, the explicit expression for $v^+(x, t)$ is given by

where $\Gamma(\xi, \tau)$ is the heat kernel given by

Consequently, the expression (2.2) gives an upper estimate for $\Psi(x, t)$ of (2.1). By considering the equation

we also give a lower estimate for $\Psi(x, t)$ of (2.1).

The following lemma gives the large time behavior of the solutions of

Lemma 2.1. (See [42,Lemma 2.4 and Remark 2.5]) Let $k > 0$ be any constant. Let $v^\pm(x, t)$ be solutions to the Cauchy problems:

Suppose that $v_0(x)$ is bounded and continuous on $\mathbb{R}^{n-2}$ and satisfies $\lim\limits_{|x|\rightarrow\infty}|v_0(x)| = 0$. Then the solutions $v^\pm(x, t)$ satisfy

respectively. Furthermore, if $v_0\in L^1(\mathbb{R}^{n-2})$, then we have

Before constructing a series of supersolutions and subsolutions, we first give some auxiliary lemmas.

Lemma 2.2. (See [10,Lemma 3.2 and Theorem 2.3]) For any $\delta\in (0, \frac{1}{2})$, there exists a positive constant $\beta: = \beta(\delta)$ such that

In addition, we have

where $m_* = \left.\sqrt{s^2-c^2}\right/c$.

Now, we introduce a lemma which plays a key role in constructing supersolutions and subsolutions.

Lemma 2.3. There exists a constant $k > 0$ such that

Proof. It follows from the first equation of (1.4) that

If we write $W = V_z$, then (2.3) becomes

Setting

then (2.4) can be rewritten as

From Lemma 2.2, there exists a constant $K > 0$ such that

By the interior $L^p$ estimates for the second derivatives of elliptic equations (see [48,Theorem 9.11]) and the embedding theorem (see [48,Theorem 7.26]), there exists a constant $\Lambda > 0$ such that

for some constant $\alpha\in (0, 1]$. Using the above estimate, the assumptions (F) and (G), and the Schauder interior estimates for the second derivatives of elliptic equations (see [48,Theorem 6.2]), we have that for some $0 < \alpha\leq1$ and $C_0 > 0$, there are

where $B_r(y_0, z_0)$ is a ball of radius $r$ in $\mathbb{R}^2$ with origin $(y_0, z_0)$. On the other hand, since $B_1(y_0, z_0)\subset\subset B_2(y_0, z_0)$ and ${\rm dist}\left(B_1\left(y_0, z_0\right), \partial B_2\left(y_0, z_0\right)\right) = 1$, we have

Here we refer the definitions of the norms $|\cdot|^*_{m, \alpha; \Omega}$, $|\cdot|_{m, \alpha; \Omega}$ and $|\cdot|_{m; \Omega}$ to ^[48] and ^[28]. Combining (2.5) and (2.6) yields

Since $W(x, z) > 0$, the Harnack-type inequality [48,Theorem 2.5] implies that there exists another constant $C_1 > 0$ such that

Taking

we obtain that

that is

This completes the proof of Lemma 2.3.

Now, we will show that the functions $V\left(y, z+v^\pm(x, t)\right)$ are a supersolution and a subsolution of (1.13), respectively. In what follows, $\Delta_x$ and $\nabla_x$ denote the $(n-2)$-dimensional Laplacian and the $(n-2)$-dimensional gradient, respectively.

Lemma 2.4. (Supersolutions and Subsolutions). Let $u(x, y, z, t)$ be the solution of (1.13) with the initial value $u_0(x, y, z)\in BUC^1(\mathbb R^n)$. Suppose that the functions $v^+(x, t)$ and $v^-(x, t)$ solving the following problems

respectively. Where $k > 0$ is the constant defined in Lemma 2.3. If

then $u^+(x, y, z, t): = V\left(y, z+v^+(x, t)\right)$ and $u^-(x, y, z, t): = V\left(y, z+v^-(x, t)\right)$ are a supersolution and a subsolution of (1.13), respectively.

Proof. We only show that $u^+(x, y, z, t)$ is a supersolution of (1.13), since the subsolution can be proved in a similar way.

Set

Using the equality $-V_{yy}-V_{zz}+\left(g'(V)+s\right)V_z-f(V) = 0$ and Lemma 2.3, we have

This completes the proof.

Now we prove Theorem 1.2.

Proof of Theorem 1.2. Let the function $v^+(x, t)$ as in Lemma 2.4. Then we have

From Lemma 2.1, we have

for some positive constant $C$. Similarly, we also obtain

This completes the proof.

Next, we give a proof of Proposition 1.3. Our argument is based again on Lemma 2.4.

Proof of Proposition 1.3. We only consider the case that $v_0\leq0, \ v_0\not\equiv0$. The right hand side inequality of (1.10) immediately follows from Theorem 1.2. To prove the left hand side inequality of (1.10), from Lemma 2.4, it suffices to show that the solution $v(x, t)$ of the problem

satisfies $v(0, t)\leq-C(1+t)^{-\frac{n-2}{2}}$ for some constant $C > 0$. Indeed, Lemma 2.4 gives that

where $C'$ is a positive constant.

Similar to (2.2), the explicit expression for $v(x, t)$ is given by

where $\Gamma(\xi, \tau)$ is the heat kernel on $\mathbb{R}^{n-2}$. Since $v_0\leq0$ and $v_0\not\equiv0$, there exists a constant $\delta > 0$ and a nonempty open set $D\subset\mathbb{R}^{n-2}$ such that $v_0\leq-\delta$ for $x\in D$. Then we have

which implies $v(0, t)\leq-C(1+t)^{-\frac{n-2}{2}}$. This completes the proof.

Now, we construct some new types of supersolutions and subsolutions.

Lemma 2.5. Let $k > 0$ be defined as in Lemma 2.3. Then there exist constants $\delta_0 > 0$, $\beta > 0$ and $\sigma\geq1$ such that, for any $\delta\in(0, \delta_0]$ and any functions $v^\pm(x, t)$ satisfying

the functions defined by

are a supersolution and a subsolution of (1.13), respectively.

Proof. Using the equality $-V_{yy}-V_{zz}+\left(s+g'(V)\right)V_z-f(V) = 0$ and Lemma 2.3, we have

On the other hand, from (F), there exists a constant $\delta_0(0 < \delta_0 < \frac{1}{8})$ such that

where

Let $\beta_3 > 0$ be defined by Lemma 2.2 with $\delta_0$. For $\delta_0\leq V\left(y, z+v^+(x, t)+\sigma\delta\left(1-e^{-\beta t}\right)\right)\leq1-\delta_0$, from (1.2) and Lemma 2.2, we have

if we take

where

For $V\left(y, z+v^+(x, t)+\sigma\delta\left(1-e^{-\beta t}\right)\right) > 1-\delta_0$ or $V\left(y, z+v^+(x, t)+\sigma\delta\left(1-e^{-\beta t}\right)\right) < \delta_0$, following from Lemma 2.2 and (2.7) we have

if we take

Thus, if we take $\beta$ small and $\sigma$ large as

we obtain $L[\tilde{u}]\geq0$. Similarly, we can obtain $L[\hat{u}]\leq0$. This completes the proof.

To prove Theorem 1.1, we need another auxiliary lemma.

Lemma 2.6. Assume that the initial value $u_0(x, y, z)\in BUC^1(\mathbb R^n)$ satisfies

Then, for any fixed $T > 0$, the solution $u(x, y, z, t)$ of (1.13) satisfies

Proof. Define a function $w(x, y, z, t)$ by

Then $\omega(x, y, z, t)$ solves the following Cauchy problem

where $\theta_i(x, y, z, t)$ is the functions that satisfy $0\leq\theta_i(x, y, z, t)\leq1$, $i = 1, 2$.

In order to prove the lemma, it suffices to consider the case where $\omega(x, y, z, 0)\geq0$ and the case where $\omega(x, y, z, 0)\leq0$, since the general case follows easily from these special cases and the comparison principle. In the following, we always assume that $\omega(x, y, z, 0)\geq0$.

Letting

then (2.10) can be rewritten as

Because of $\omega(x, y, z, 0)\geq0$, the maximum principle gives $\omega(x, y, z, t)\geq0$. Then, by assumptions (F) and (G), there exists a constant $M_* > 0$ such that

Since $g\in C^{2+\gamma_0}(\mathbb{R})$ and $g_1(x, y, z, t)$ is bounded continuous in $\mathbb{R}^n\times\mathbb{R}^+$, Friedman [49,Chapter 9,Theorem 2] implies that the fundamental solution $\Gamma(x, y, z, \zeta, \eta_1, \eta_2, t, \tau)$ of the problem

exists and satisfies

where $c_1$, $c_2$ are positive constants depending only on $T$. A solution of problem (2.11) can be expressed as

Then we obtain the estimate

Since (2.9), then for any fixed $T > 0$, we have

Similarly, we can treat the case that $\omega(x, y, x, 0)\leq0$. This completes the proof.

Next, we end this section by proving Theorem 1.1.

Proof of Theorem 1.1. We only show a lower estimate, since an upper estimate is obtained similarly. Taking constants $k > 0$ as in Lemma 2.3 and $\sigma\geq1$ as in Lemma 2.5. Let a constant $\varepsilon > 0$ be arbitrarily fixed. Define a constant $\hat{\varepsilon}: = \varepsilon/\left(2\|V_z\|_{L^\infty(\mathbb{R}^2)}+1\right)$. Since $f(r) > 0$ for $r < 0$ by the assumption (F), we have that

by the comparison principle. Consequently, there exists a constant $T_1 > 0$ such that

Furthermore, Lemma 2.6 implies that there exists a constant $R > 0$ that satisfies

Then we can take a function $v_0(x)\geq0$ that satisfies $\lim_{|x|\rightarrow\infty}v_0(x) = 0$ and

Assume that $v(x, t)$ solves the following Cauchy problem:

Then Lemma 2.1 implies that there exists a constant $T_2 > 0$ for which $v(x, t)\leq\hat{\varepsilon}$ holds true for $t\geq T_2$. Finally, by using the comparison principle and the solution constructed in Lemma 2.5, we obtain

for $t\geq T_1+T_2$. This completes the proof of Theorem 1.1.

3.   Stability and unstability of V-shaped fronts in Case B

In this section, we consider Case B and give proofs of Theorems 1.5 and 1.6.

Proof of Theorem 1.5. Consider the following two-dimensional problem

Denote the solution by $u(y, z, t; \hat{u}_0)$. Following from (1.6) (see also ^[10]), we have that

It is clear that $u(y, z, t; \hat{u}_0)$ is also a solution of (1.1) with initial value $u(x, y, z, 0) = \hat{u}_0(y, z)$. Since $V(y, z)\leq u_0(x, y, z)\leq \hat{u}_0(y, z)$ in $(x, y, z)\in\mathbb{R}^3$, it follows from the comparison principle that

where $u(x, y, z, t; u_0)$ denotes the solution of (1.1). Finally, using (3.2) and (3.3) we obtain

This completes the proof.

We now give the proof of Theorem 1.6. This theorem implies that the V-shaped traveling fronts are not necessarily asymptotically stable if the initial perturbations are not decay to zero as $|x|+|y|+|z|\rightarrow\infty$ even they are very small. By using supersolutions and subsolutions constructed in the previous section and the following lemma, we construct a sequence of supersolutions and subsolutions that push the solution back and forth in the $z$-direction, then forcing the solution to oscillate permanently with non-decaying amplitude.

Lemma 3.1. (See [42,Lemma 3.1 and Lemma 3.2]). Let $k > 0$ be defined as in Lemma 2.3 and $v^\pm(x, t)$ be the solutions to the problem

respectively. Suppose that the initial value $v^\pm_0(x)$ are bounded on $\mathbb R$ and satisfy

and

for some constant $\delta > 0$ and some integer $m\geq2$, respectively. Then there exists a constant $C > 0$ depending only on $\delta$ and $k$ such that

and

respectively, where $T = m(m!)^2/4$.

Proof of Theorem 1.6. Set

Define two sequences of smooth functions $\left\{v^\pm_{0, i}(x)\right\}_{i = 1, 2, \cdots}$ satisfying

and

respectively. We also take a function $v_0^*(x)\in C^\infty(\mathbb R)$ to satisfy

Let $u^*(x, y, z, t)$ be the solution to (1.13) with $u^*(x, y, z, 0) = V\left(y, z+v^*_0(x)\right)$ and $v^+_i(x, t)$ be the solution to the following Cauchy problem:

From $-\delta\leq v^*_0(x)\leq v^+_{0, i}(x)$, we have

then Lemma 2.4 implies that

Thus, it follows from Lemma 3.1 that

where $t_{2i} = (2i)\left((2i)!\right)^2/4$. This implies that

Similarly, again by using Lemma 3.1 and the equalities $v^-_{0, i}(x)\leq v^*_0(x)\leq\delta$ for $i = 1, 2, 3, \cdots$, we get

where $t_{2i+1} = (2i+1)\left((2i+1)!\right)^2/4$. Then, we have

Combining (3.4) and (3.5), we obtain the expected result. This completes the proof.

4.   Discussion

Recently, the study on multidimensional traveling fronts for scalar reaction-diffusion equations has attracted much attention. For example, V-formed curved fronts for two-dimensional spaces (see ^{[10,11,12,15,16,17,20,28,33]}), cylindrically symmetric traveling fronts (see ^[14,30,36]) and traveling fronts with pyramidal shapes (see ^{[21,23,24,25,35,37,45]}) in higher-dimensional spaces. For reaction-diffusion system, we refer to ^{[26,27,29,31,32]}.

In ^[10], under the assumptions (F) and (G), we establish the existence and stability of V-shaped traveling fronts $V(y, z)$ of (1.1) in $\mathbb{R}^2$ for every direction $\theta\in (0, \pi/2)$ satisfying (C). On the basis of ^[10], we establish the multidimensional asymptotic stability of V-shaped traveling fronts $V(y, z)$ in $\mathbb{R}^n (n\geqslant3)$ in this article. Here we would like to mention that the main method of this paper comes from Matano et al.^[42] and Sheng et al.^[28]. However, in order to overcome the difficulties caused by nonlinear convection, we have to choose a suitable space for the initial data $u_0$, namely $u_0\in C^1(\mathbb{R}^2)$. Moreover, there seems to be no research on the multidimensional asymptotic stability of the nonplanar traveling fronts for a reaction-diffusion equation with nonlinear convection, even if one-dimensional traveling fronts (or planar traveling fronts).

Acknowledgments

The authors would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper. This work was supported by the NSF of China (11701012).

Conflict of interest

The authors declare no conflict of interest.