Global and blow up solutions to cross diffusion systems on 3D domains
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- by Dung Le and Vu Thanh Nguyen
- Proc. Amer. Math. Soc. 144 (2016), 4845-4859
- DOI: https://doi.org/10.1090/proc/13102
- Published electronically: April 20, 2016
- PDF | 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
Bibliographic Information
- Dung Le
- Affiliation: Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249
- MR Author ID: 367842
- Email: dle@math.utsa.edu
- Vu Thanh Nguyen
- Affiliation: Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249
- MR Author ID: 758903
- Email: vu.nguyen@utsa.edu
- Received by editor(s): November 2, 2014
- Received by editor(s) in revised form: January 16, 2016
- Published electronically: April 20, 2016
- Additional Notes: The second author was partially supported by NSF grant DMS0707229
- Communicated by: Catherine Sulem
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4845-4859
- MSC (2010): Primary 35J70, 35B65; Secondary 42B37
- DOI: https://doi.org/10.1090/proc/13102
- MathSciNet review: 3544534