This paper is concerned with the nonlocal dispersal logistic equation $\left\{\begin{array}{cc}J\ast u\left(x\right)-u\left(x\right)=-\lambda u\left(x\right)+a\left(x\right)u^p\hfill & \text{ in }\bar{\Omega },\hfill \\ u\left(x\right)=0\hfill & \text{ in }{\mathbb{R}}^N\setminus \bar{\Omega },\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^N$ $(N\ge 1)$ is a bounded domain, $\lambda$ and $p>1$ are constants, the dispersal kernel $J$ and the coefficient $a\left(x\right)$ are nonnegative. The work of García-Melián and Rossi (2009) reveals that the nonlocality makes a key change of positive solutions. In this paper, we investigate the sharp effects of degeneracy on the behavior of positive solutions. We find that the effect is different from the diffusive logistic equation.

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简并性对非局部色散逻辑方程的影响

本文涉及非局部扩散对数方程 $\left\{\begin{array}{cc}Ĵ\ast ü（X）-ü（X）=-\lambda ü（X）+\mathrm{一种}（X）{ü}^p\hfill & \text{ 在 }\bar{\Omega }，\hfill \\ ü（X）=0\hfill & \text{ 在 }{\mathbb{[R}}^{ñ}\setminus \bar{\Omega }，\hfill \end{array}\right.$在哪里 $\Omega \subset {\mathbb{[R}}^{ñ}$ $（ñ\ge \mathrm{1个}）$ 是一个有界域， $\lambda$ 和 $p>\mathrm{1个}$ 是常数，分散核 $Ĵ$ 和系数 $\mathrm{一种}（X）$是非负的。García-Melián和Rossi（2009）的工作表明，非本地性是积极解决方案的关键变化。在本文中，我们研究了退化对正解行为的尖锐影响。我们发现效果与扩散对数方程不同。