Global well-posedness of a conservative relaxed cross diffusion system

SIAM Journal on Mathematical Analysis

Volumn 44, Issue 3, 2012, Pages 1674-1693

  Lepoutre, Thomas ^a   Pierre, Michel ^b   Rolland, Guillaume ^b  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Cross diffusion; Global existence; Population dynamics; Quasilinear systems; Reaction diffusion

Indexed keywords

CROSS-DIFFUSION; CROSS-DIFFUSION SYSTEM; GLOBAL EXISTENCE; GLOBAL WEAK SOLUTION; GLOBAL WELL-POSEDNESS; LIPSCHITZ CONTINUOUS; NONLINEAR OPERATOR; QUASILINEAR SYSTEM; REACTION DIFFUSION; SPACE DIMENSIONS;

DIFFUSION; POPULATION DYNAMICS;

CONTINUOUS TIME SYSTEMS;

EID: 84861403589     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/110848839     Document Type: Article

References (29)

1. B. Andreianov, M. Bendahmane, and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math. Models Methods Appl. Sci., 21 (2011), pp. 307-344.
2. M. Bendahmane and M. Langlais, A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Evol. Equ., 10 (2010), pp. 883-904.
3. M. Bendahmane, T. Lepoutre, A. Marrocco, and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, J. Math. Pures Appl. (9), 92 (2009), pp. 651-667.
4. M. Bendahmane and M. Sepulveda, Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), pp. 823-853.
5. H. Berestycki, G. Nadin, B. Perthame, and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), pp. 2813-2844.
6. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
7. L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), pp. 39-59. (Pubitemid 43585103)
8. S. Genieys, V. Volpert, and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Mod. Nat. Phenom., 1 (2006), pp. 65-82.
9. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, New York, 1977.
10. D. Guangchang, Initial and nonlinear oblique boundary value problem for fully nonlinear parabolic equations, J. Partial Differential Equations, 1 (1988), pp. 12-42.
11. M. Iida, M. Mimura, and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), pp. 617-641. (Pubitemid 44326894)
12. A. Jüngel, Diffusive and nondiffusive population models, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, G. Naldi, L. Paresch, and G. Toscani, eds. Birkhäuser Boston, Boston, 2010.
13. N. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Math. Appl. (Soviet Ser.) 7, D. Reidel Publishing, Dordrecht, The Netherlands, 1987.
14. G. Ladyzhenskaya, V. Solonnikov, and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., 23, American Mathematical Society, Providence, RI, 1968.
15. T. Lepoutre, Analyse et mod́elisation de ph́enom̀enes de croissance et mouvement issus de la biologie, Ph.D. thesis, Universit́e Pierre et Marie Curie, Paris, 2009.
16. T. Lepoutre and S. Martinez, Steady States of Relaxed Cross Diffusion Models, in preparation.
17. S. Levin and L. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976), p. 659.
18. S. A. Levin, Dispersion and Population Interactions, Amer. Naturalist, 108 (1974), pp. 207-228.
19. Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains, Discrete Contin. Dyn. Syst., 12 (2005), pp. 185-192. (Pubitemid 40419368)
20. G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, 2005.
21. A. Maugeri, D. Palagachev, and L. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Math. Res. 109, Wiley-VCH, Berlin, 2000.
22. M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), pp. 49-64. (Pubitemid 10116367)
23. M. Mimura and J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, J. Theoret. Biol., 75 (1978), pp. 249-262. (Pubitemid 9105427)
24. M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys., 15 (1982), pp. 19-65.
25. G. Nadin, B. Perthame, and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), pp. 553-557.
26. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics 10, Springer-Verlag, Berlin, New York, 1980.
27. M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), pp. 417-455.
28. N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), pp. 83-99.
29. Y. Wang, The global existence of solutions for a cross-diffusion system, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), pp. 519-528. (Pubitemid 41060081)