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2006, 14(1): 93-116. doi: 10.3934/dcds.2006.14.93

Multiple stable patterns for some reaction-diffusion equation in disrupted environments

1. 

Aisin AW. Co. Ltd., Q21 Promotion Department, Engineering Division, 0 Takane, Fujii-Cho, Anjo, Aichi, 444-1192, Japan

2. 

Department of Mathematics and Infomation Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi,Tokyo 192-0397, Japan

3. 

Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjyuku-ku, Tokyo 169-8555, Japan

Received  November 2004 Revised  March 2005 Published  October 2005

We study the existence of multiple positive stable solutions for

$ -\epsilon^2\Delta u(x) = u(x)^2(b(x)-u(x)) \ \mbox{in}\ \Omega, \quad$ $ \frac{\partial u}{\partial n}(x) = 0 \ \mbox{on}\ \partial\Omega.$

Here $\epsilon>0$ is a small parameter and $b(x)$ is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

Citation: Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka. Multiple stable patterns for some reaction-diffusion equation in disrupted environments. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 93-116. doi: 10.3934/dcds.2006.14.93
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