Global and blow up solutions to cross diffusion systems on 3D domains

Le, Dung; Nguyen, Vu

doi:10.1090/proc/13102

Global and blow up solutions to cross diffusion systems on 3D domains
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by Dung Le and Vu Thanh Nguyen PDF

Proc. Amer. Math. Soc. 144 (2016), 4845-4859 Request permission

Abstract:

Necessary and sufficient conditions for global existence of classical solutions to a class of cross diffusion systems on 3-dimensional domains are studied. Examples of blow up solutions are also given.

References

Dung Le and Vu Thanh Nguyen, Global solutions to cross diffusion parabolic systems on 2D domains, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2999–3010. MR 3336624, DOI 10.1090/S0002-9939-2015-12501-4
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
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933, DOI 10.1142/9789812795557
Konrad Horst Wilhelm Küfner, Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model, NoDEA Nonlinear Differential Equations Appl. 3 (1996), no. 4, 421–444. MR 1418589, DOI 10.1007/BF01193829
Le Dung, Global attractors and steady state solutions for a class of reaction-diffusion systems, J. Differential Equations 147 (1998), no. 1, 1–29. MR 1632736, DOI 10.1006/jdeq.1998.3435
Dung Le, Cross diffusion systems on $n$ spatial dimensional domains, Indiana Univ. Math. J. 51 (2002), no. 3, 625–643. MR 1911048, DOI 10.1512/iumj.2002.51.2198
Le Dung and Hal L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor, J. Differential Equations 130 (1996), no. 1, 59–91. MR 1409023, DOI 10.1006/jdeq.1996.0132
D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., Vol. 56, No. 4, pp. 1749–1791, 2007.
Yuan Lou and Wei-Ming Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996), no. 1, 79–131. MR 1415047, DOI 10.1006/jdeq.1996.0157
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
Zhuangyi Liu and Songmu Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1681343