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
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by Phan Văn Tuộc

Proc. Amer. Math. Soc. 135 (2007), 3933-3941

DOI: https://doi.org/10.1090/S0002-9939-07-08978-2

Published electronically: August 2, 2007

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Abstract:

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in $n$-dimensional domains ($n\geq 1$). Global existence of classical solutions under certain restrictions on the coefficients is established.

References

Herbert Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal. 12 (1988), no. 9, 895–919. MR 960634, DOI 10.1016/0362-546X(88)90073-9
Herbert Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), no. 1, 13–75. MR 1014726
Herbert Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z. 202 (1989), no. 2, 219–250. MR 1013086, DOI 10.1007/BF01215256
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
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, DOI 10.3934/dcds.2003.9.1193
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, DOI 10.3934/dcds.2004.10.719
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
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. MR 2137864, DOI 10.1090/S0002-9939-05-07867-6
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, DOI 10.3934/dcds.2005.12.185
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, DOI 10.3934/dcds.1998.4.193
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, DOI 10.1007/978-1-4612-5282-5
Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83–99. MR 540951, DOI 10.1016/0022-5193(79)90258-3
J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983).