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Steady state analysis for a relaxed cross diffusion model

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  • 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.
    Mathematics Subject Classification: Primary: 35K55, 35B32, 35B35.

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