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

do Nascimento, Arnaldo Simal

Electronic Journal of Differential Equations, Tome 1997 (1997).

Résumé

Summary: We use $$ \eqalign{ {\partial v_\varepsilon \over \partial t} =\ \varepsilon^2\, \hbox{div}\, (k_1(x)\nabla v_\varepsilon) + k_2(x)(v_\varepsilon -\alpha)(\beta-v_\varepsilon) (v_\varepsilon -\gamma_\varepsilon(x))\,,\hbox{ in }\Omega\times{\Bbb R}^+ \cr \ v_\varepsilon(x,0) = v_0 \quad {\partial v_\varepsilon \over \partial \widehat{n}} = 0\,, \quad\hbox{ for } x\in \partial\Omega\,, t\ positive\,.} $$ 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.

Détail

Classification : 35K20, 35K57, 35B25
Keywords: diffusion equation, gamma-convergence, transition layers, stable equilibria