A nonlocal dispersal logistic equation with spatial degeneracy

Jian-Wen Sun; Wan-Tong Li; Zhi-Cheng Wang

doi:10.3934/dcds.2015.35.3217

Article Contents

2015, Volume 35, Issue 7: 3217-3238. Doi: 10.3934/dcds.2015.35.3217

This issue Previous Article Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration Next Article Isolated singularity for semilinear elliptic equations

A nonlocal dispersal logistic equation with spatial degeneracy

1.
School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received: March 31, 2014

Revised: October 31, 2014

Published: May 2017

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Abstract

In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.

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Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 92D25.

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