MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION

Abstract

In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$-dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\mathbb{R}^n )$ in the form of ${\rm{e}}^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$($\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\mathbb{R}^n)$ space and $H^k(\mathbb{R}^n) (k \geq [\frac{n+1}{2}])$ space.