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

Takanori Ide; Kazuhiro Kurata; Kazunaga Tanaka

doi:10.3934/dcds.2006.14.93

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

Takanori Ide^*, Kazuhiro Kurata, Kazunaga Tanaka

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

We study the existence of multiple positive stable solutions for -ε²Δu(x) = u(x)²(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 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.

Original language	English
Pages (from-to)	93-116
Number of pages	24
Journal	Discrete and Continuous Dynamical Systems
Volume	14
Issue number	1
DOIs	https://doi.org/10.3934/dcds.2006.14.93
Publication status	Published - 2006 Jan

Keywords

Nonlinear elliptic equations
Pattern formation
Singular perturbation
Stable solutions
Variational methods

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2006.14.93

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abstract = "We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 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.",

keywords = "Nonlinear elliptic equations, Pattern formation, Singular perturbation, Stable solutions, Variational methods",

author = "Takanori Ide and Kazuhiro Kurata and Kazunaga Tanaka",

year = "2006",

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AU - Tanaka, Kazunaga

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AB - We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 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.

KW - Nonlinear elliptic equations

KW - Pattern formation

KW - Singular perturbation

KW - Stable solutions

KW - Variational methods

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

Abstract

Keywords

ASJC Scopus subject areas

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