This paper is concerned with the multidimensional stability of V-shaped traveling fronts for a reaction-diffusion equation with nonlinear convection term in $\mathbb{R}^n$ ($n\geq3$). We consider two cases for initial perturbations: one is that the initial perturbations decay at space infinity and another one is that the initial perturbations do not necessarily decay at space infinity. In the first case, we show that the V-shaped traveling fronts are asymptotically stable. In the second case, we first show that the V-shaped traveling fronts are also asymptotically stable under some further assumptions. At the same time, we also show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which means that the traveling fronts are not asymptotically stable under general bounded perturbations.

中文翻译：

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具有非线性对流的双稳态反应扩散方程中V形行进面的多维稳定性

本文关注具有$ \ mathbb {R} ^ n $（$ n \ geq3 $）中具有非线性对流项的反应扩散方程的V形行进面的多维稳定性。对于初始扰动，我们考虑两种情况：一种是初始扰动在空间无限处衰减，另一种情况是初始扰动不一定在空间无限处衰减。在第一种情况下，我们表明V形行进线是渐近稳定的。在第二种情况下，我们首先证明在某些进一步的假设下，V形行进线也渐近稳定。同时，我们还表明，存在一个在两个V形行进前沿之间永久振动的解决方案，这意味着行进前沿在一般有界扰动下不是渐近稳定的。