Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions

Tuộc, Phan

doi:10.1090/S0002-9939-07-08978-2

Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions

Author: Phan Van Tuôc
Journal: Proc. Amer. Math. Soc. 135 (2007), 3933-3941
MSC (2000): Primary 35B50, 35K50, 35K55, 35K57
Published electronically: August 2, 2007
MathSciNet review: 2341943
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Abstract: We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in -dimensional domains ( $n\geq 1$ ). Global existence of classical solutions under certain restrictions on the coefficients is established.

1. Herbert Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal. 12 (1988), no. 9, 895–919. MR 960634, 10.1016/0362-546X(88)90073-9
2. Herbert Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), no. 1, 13–75. MR 1014726
3. Herbert Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z. 202 (1989), no. 2, 219–250. MR 1013086, 10.1007/BF01215256
4. H. Amann, Abstract methods in differential equations, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 1, 89–105 (English, with English and Spanish summaries). MR 2037227
5. Y. S. Choi, Roger Lui, and Yoshio Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1193–1200. MR 1974423, 10.3934/dcds.2003.9.1193
6. Y. S. Choi, Roger Lui, and Yoshio Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 719–730. MR 2018876, 10.3934/dcds.2004.10.719
7. Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384
8. Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
9. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
10. Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184
11. Dung Le and Toan Trong Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension, Proc. Amer. Math. Soc. 133 (2005), no. 7, 1985–1992 (electronic). MR 2137864, 10.1090/S0002-9939-05-07867-6
12. Yi Li and Chunshan Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains, Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 185–192. MR 2122161
13. Yuan Lou, Wei-Ming Ni, and Yaping Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Systems 4 (1998), no. 2, 193–203. MR 1616969, 10.3934/dcds.1998.4.193
14. Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825
15. Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83–99. MR 540951, 10.1016/0022-5193(79)90258-3
16. J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983).