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

Takanori Ide; Kazuhiro Kurata; Kazunaga Tanaka

doi:10.3934/dcds.2006.14.93

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2006, Volume 14, Issue 1: 93-116. Doi: 10.3934/dcds.2006.14.93

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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
2.
Department of Mathematics and Infomation Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi,Tokyo 192-0397
3.
Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjyuku-ku, Tokyo 169-8555

Received: October 31, 2004

Revised: February 28, 2005

Published: February 2006

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Abstract

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.

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Mathematics Subject Classification: Primary 35J65, 35B05, 58E05; Secondary: 92C15.

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