Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Steady state analysis for a relaxed cross diffusion model

Pages: 613 - 633, Volume 34, Issue 2, February 2014      doi:10.3934/dcds.2014.34.613

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Thomas Lepoutre - INRIA Rhône Alpes (team DRACULA), Batiment CEI-1, 66 Boulevard NIELS BOHR, 69603 Villeurbanne cedex, France (email)
Salomé Martínez - Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, $5^o $ piso-Santiago, Chile (email)

Abstract: In this article we study the existence the existence of nonconstant steady state solutions for the following relaxed cross-diffusion system $$ \left\lbrace\begin{array}{l} \partial_t u-\Delta[a(\tilde v)u]=0,\;\text{ in } (0,\infty)\times\Omega,\\ \partial_t v-\Delta[b(\tilde u)v]=0,\;\text{ in } (0,\infty)\times\Omega,\\ -\delta\Delta \tilde u+\tilde u=u,\;\text{ in }\Omega,\\ -\delta\Delta \tilde v+\tilde v=v,\;\text{ in }\Omega,\\ \partial_n u=\partial_n v=\partial\tilde u=\partial_n\tilde u=0,\;\text{ on } (0,\infty) \times \partial\Omega, \end{array}\right. $$ with $\Omega$ a bounded smooth domain, $n$ the outer unit normal to $\partial\Omega$, $\delta>0$ denotes the relaxation parameter. The functions $a(\tilde v)$, $b(\tilde u)$ account for nonlinear cross-diffusion, being $a(\tilde v)=1+{\tilde v}^\gamma$, $b(\tilde u)=1+{\tilde u}^\eta$ with $\gamma, \eta >1$ a model example. We give conditions for the stability of constant steady state solutions and we prove that under suitable conditions Turing patterns arise considering $\delta$ as a bifurcation parameter.

Keywords:  Cross diffusion models, bifurcation analysis, stability analysis, duality estimates.
Mathematics Subject Classification:  Primary: 35K55, 35B32, 35B35.

Received: September 2012;      Revised: April 2013;      Available Online: August 2013.