Abstract

In this article we study the existence the existence of nonconstant steady state solutions for the following relaxed cross-diffusion system {partial derivative(t)u - Delta[a((v) over tildeu] = 0, in (0, infinity) x Omega, partial derivative(t)v - Delta[b((u) over tildev] = 0, in (0, infinity) x Omega, -delta Delta(u) over tilde + (u) over tilde = u, in Omega, -delta Delta(v) over tilde + (v) over tilde = u, in Omega, partial derivative(n)u = partial derivative(n)v = partial derivative(u) over tilde = partial derivative(n)(u) over tilde = 0, on (0, infinity) x partial derivative Omega, with Omega a bounded smooth domain, n the outer unit normal to partial derivative Omega, delta> 0 denotes the relaxation parameter. The functions a((v) over tilde)), b((u) over tilde) account for nonlinear crossdiffusion, being a((v) over tilde) = 1+ (v) over tilde gamma, b (u) over tilde = 1+(u) over tilde eta with gamma, n > 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 as a bifurcation parameter.