Steady state analysis for a relaxed cross diffusion model

Thomas Lepoutre 1, 2 Salome Martinez 3, 4
1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
ICJ - Institut Camille Jordan [Villeurbanne], Inria Grenoble - Rhône-Alpes, CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire
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.
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Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 34 (2), pp.613-633. 〈10.3934/dcds.2014.34.613〉
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Contributeur : Thomas Lepoutre <>
Soumis le : mardi 3 septembre 2013 - 15:35:59
Dernière modification le : mercredi 11 avril 2018 - 01:57:35

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

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