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Journal Articles Discrete and Continuous Dynamical Systems - Series A Year : 2013

Steady state analysis for a relaxed cross diffusion model

Abstract

In this article we study the existence the existence of nonconstant steady state solutions for the following relaxed cross-diffusion system ⎧∂tu−Δ[a(v˜)u]=0, in (0,∞)×Ω,∂tv−Δ[b(u˜)v]=0, in (0,∞)×Ω,−δΔu˜+u˜=u, in Ω,−δΔv˜+v˜=v, in Ω,∂nu=∂nv=∂u˜=∂nu˜=0, on (0,∞)×∂Ω, with Ω a bounded smooth domain, n the outer unit normal to ∂Ω, δ>0 denotes the relaxation parameter. The functions a(v˜), b(u˜) account for nonlinear cross-diffusion, being a(v˜)=1+v˜γ, b(u˜)=1+u˜η with γ,η>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 δ as a bifurcation parameter.

Dates and versions

hal-00857467 , version 1 (03-09-2013)

Identifiers

Cite

Thomas Lepoutre, Salome Martinez. Steady state analysis for a relaxed cross diffusion model. Discrete and Continuous Dynamical Systems - Series A, 2013, 34 (2), pp.613-633. ⟨10.3934/dcds.2014.34.613⟩. ⟨hal-00857467⟩
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