# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 13-75. [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), 651-667. 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), 39-59. 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), 321-340. 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. [6] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., 7 (2007), 491-511. [7] P. Deuring, An initial-boundary-value problem for a certain density-dependent diffusion system, Mathematische Zeitschrift, 194 (1987), 375-396. 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), 315-347. [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), 1674-1693. doi: 10.1137/110848839. [10] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035. [11] 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. [12] 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. [13] 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. [14] 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.

show all references

##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [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), 651-667. 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), 39-59. 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), 321-340. 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. [6] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., 7 (2007), 491-511. [7] P. Deuring, An initial-boundary-value problem for a certain density-dependent diffusion system, Mathematische Zeitschrift, 194 (1987), 375-396. 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), 315-347. [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), 1674-1693. doi: 10.1137/110848839. [10] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035. [11] 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. [12] 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. [13] 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. [14] 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.
 [1] Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133 [2] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [3] Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i [4] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [5] Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785 [6] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [7] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [8] Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3073-3081. doi: 10.3934/dcdss.2020135 [9] Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635 [10] Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Global stability analysis for SEIS models with n latent classes. Mathematical Biosciences & Engineering, 2008, 5 (1) : 20-33. doi: 10.3934/mbe.2008.5.20 [11] Matteo Piu, Gabriella Puppo. Stability analysis of microscopic models for traffic flow with lane changing. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022006 [12] Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 [13] Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244 [14] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 [15] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [16] Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745 [17] Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29 [18] Kuo-Shou Chiu. Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 659-689. doi: 10.3934/dcdsb.2021060 [19] Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324 [20] Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069

2020 Impact Factor: 1.392