Electron. J. Diff. Eqns., Vol. 1997(1997), No. 22, pp 1-17.

Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains

Arnaldo Simal do Nascimento

Abstract:
We use $\Gamma$-convergence to prove existence of stable multiple-layer stationary solutions (stable patterns) to the reaction-diffusion equation.
$$ \eqalign{ {\partial v_\varepsilon \over \partial t} =& \varepsilon^2\, \hbox{div}\, (k_1(x)\nabla v_\varepsilon) + k_2(x)(v_\varepsilon -\alpha)(\beta-v_\varepsilon) (v_\varepsilon -\gamma_\varepsilon(x))\,,\hbox{ in }\Omega\times{\Bbb R}^+ \cr & v_\varepsilon(x,0) = v_0 \quad {\partial v_\varepsilon \over \partial \widehat{n}} = 0\,, \quad\hbox{ for } x\in \partial\Omega\,, t\ positive\,.} $$
Given nested simple closed curves in ${\Bbb R}^2$, we give sufficient conditions on their curvature so that the reaction--diffusion problem possesses a family of stable patterns. In particular, we extend to two-dimensional domains and to a spatially inhomogeneous source term, a previous result by Yanagida and Miyata.

Submitted May 13, 1997. Published December 1, 1997.
Math Subject Classification: 35K20, 35K57, 35B25.
Key Words: Diffusion equation, Gamma-convergence, transition layers, stable equilibria.

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Arnaldo Simal do Nascimento
Universidade Federal de Sao Carlos, D.M.; 13565-905 - Sao Carlos, S.P. Brazil
e-mail: dasn@power.ufscar.br
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