    February  2014, 34(2): 613-633. doi: 10.3934/dcds.2014.34.613

## Steady state analysis for a relaxed cross diffusion model

 1 INRIA Rhône Alpes (team DRACULA), Batiment CEI-1, 66 Boulevard NIELS BOHR, 69603 Villeurbanne cedex, France 2 Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, $5^o$ piso-Santiago, Chile

Received  September 2012 Revised  April 2013 Published  August 2013

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.
Citation: Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613
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##### References:
  H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. Google Scholar  M. Bendahmane, T. Lepoutre, A. Marrocco and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, Journal de Mathmatiques Pures et Appliques (9), 92 (2009), 651-667. doi: 10.1016/j.matpur.2009.05.003.  Google Scholar  L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002.  Google Scholar  M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar  L. Desvillettes and F. Conforto, Rigorous passage to the limit in a system of reaction-diffusion equations towards a system including cross diffusions, CMLA2009-34, 2009. Google Scholar  L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., 7 (2007), 491-511. Google Scholar  P. Deuring, An initial-boundary-value problem for a certain density-dependent diffusion system, Mathematische Zeitschrift, 194 (1987), 375-396. doi: 10.1007/BF01162244.  Google Scholar  H. Izuhara and M. Mimura, Reaction-diffusion system approximation to the cross-diffusion competition system, Hiroshima Math. J., 38 (2008), 315-347. Google Scholar  T. Lepoutre, M. Pierre and G. Rolland, Global well-posedness of a conservative relaxed cross diffusion system, SIAM Journal on Mathematical Analysis, 44 (2012), 1674-1693. doi: 10.1137/110848839.  Google Scholar  M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035.  Google Scholar  L. Nirenberg, Topics in nonlinear functional analysis, Chapter 6 by E. Zehnder, Notes by R. A. Artino, Revised reprint of the 1974 original, Courant Lecture Notes in Mathematics, 6, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. Google Scholar  M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106 (electronic). doi: 10.1137/S0036144599359735.  Google Scholar  N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.  Google Scholar  Y. Wang, The global existence of solutions for a cross-diffusion system, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), 519-528. doi: 10.1007/s10255-005-0260-9.  Google Scholar
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