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 - A, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.

[2]

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. doi: 10.1016/j.matpur.2009.05.003.

[3]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39. doi: 10.1016/j.jde.2005.08.002.

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2.

[5]

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), 2009.

[6]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion,, Adv. Nonlinear Stud., 7 (2007), 491.

[7]

P. Deuring, An initial-boundary-value problem for a certain density-dependent diffusion system,, Mathematische Zeitschrift, 194 (1987), 375. doi: 10.1007/BF01162244.

[8]

H. Izuhara and M. Mimura, Reaction-diffusion system approximation to the cross-diffusion competition system,, Hiroshima Math. J., 38 (2008), 315.

[9]

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. doi: 10.1137/110848839.

[10]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035.

[11]

L. Nirenberg, Topics in nonlinear functional analysis,, Chapter 6 by E. Zehnder, 6 (1974).

[12]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Rev., 42 (2000), 93. doi: 10.1137/S0036144599359735.

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3.

[14]

Y. Wang, The global existence of solutions for a cross-diffusion system,, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), 519. doi: 10.1007/s10255-005-0260-9.

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.

[2]

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. doi: 10.1016/j.matpur.2009.05.003.

[3]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39. doi: 10.1016/j.jde.2005.08.002.

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2.

[5]

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), 2009.

[6]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion,, Adv. Nonlinear Stud., 7 (2007), 491.

[7]

P. Deuring, An initial-boundary-value problem for a certain density-dependent diffusion system,, Mathematische Zeitschrift, 194 (1987), 375. doi: 10.1007/BF01162244.

[8]

H. Izuhara and M. Mimura, Reaction-diffusion system approximation to the cross-diffusion competition system,, Hiroshima Math. J., 38 (2008), 315.

[9]

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. doi: 10.1137/110848839.

[10]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035.

[11]

L. Nirenberg, Topics in nonlinear functional analysis,, Chapter 6 by E. Zehnder, 6 (1974).

[12]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass,, SIAM Rev., 42 (2000), 93. doi: 10.1137/S0036144599359735.

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3.

[14]

Y. Wang, The global existence of solutions for a cross-diffusion system,, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), 519. doi: 10.1007/s10255-005-0260-9.

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