Electronic Journal of Differential Equations,
Vol. 1997(1997), No. 22, pp 1-17.
Title: Stable multiple-layer stationary solutions of a semilinear parabolic
equation in two-dimensional domains
Author: Arnaldo Simal do Nascimento (Univ. Federal de Sao Carlos, Brazil)
Abstract:
We use $\Gamma$--convergence to prove existence of stable multiple--layer
stationary solutions (stable patterns) to the reaction--diffusion equation.
$$ \eqalign{
{\partial v_\varepsilon \over \partial t} =& \varepsilon^2\, \hbox{div}\,
(k_1(x)\nabla v_\varepsilon) + k_2(x)(v_\varepsilon -\alpha)(\beta-v_\varepsilon)
(v_\varepsilon -\gamma_\varepsilon(x))\,,\hbox{ in }\Omega\times{\Bbb R}^+ \cr
&v_\varepsilon(x,0) = v_0 \quad
{\partial v_\varepsilon \over \partial \widehat{n}} = 0\,, \quad\hbox{ for }
x\in \partial\Omega\,, \ t >0\,.}
$$
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.