On a time-dependent bio-reactor model with chemotaxis

Applied Mathematics and Computation

Volumn 131, Issue 2-3, 2002, Pages 531-558

  Dung, Le ^a  

^a UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

Author keywords

Cross diffusion system; Global attractor; Index theory; Periodic solutions

Indexed keywords

EID: 0037174274     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0096-3003(01)00168-0     Document Type: Article

References (30)

1. R.A. Adams Sobolev Spaces 1975 Academic Press New York
2. H. Amann Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations Nonlinear Anal., T.M.A. 12 1988 895 919
3. H. Amann Dynamic theory of quasilinear parabolic systems-III. Global existence Math. Z. 202 1989 219 250
4. H. Amann Dynamic theory of quasilinear parabolic equations-II. Reaction–diffusion systems Diff. Int. Eqs. 3 1990 13 75
5. F. Kelly K. Dapsis D. Lauffenburger Effect of bacterial chemotaxis on dynamics of microbial competition Microbial. Ecol. 16 1988 115 131
6. D. Lauffenburger R. Aris K. Keller Effects of cell motility and chemotaxis on microbial population growth Biophys. J. 40 1982 209 219
7. D. Lauffenburger Quantitative studies of bacterial chemotaxis and microbial population dynamics Microbial. Ecol. 22 1991 175 185
8. L. Dung Coexistence with chemotaxis SIAM Math. Anal. 32 3 2000 504 521
9. L. Dung Global attractors and steady state solutions for a class of reaction diffusion systems J. Diff. Eqs. 147 1998 1 29
10. L. Dung Hölder regularity for certain strongly coupled parabolic systems J. Diff. Eqs. 151 1999 313 344
11. L. Dung Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems Nonlinear Anal., T.M.A. 39 2000 157 171
12. L. Dung Dissipativity and global attractor for a class of quasilinear parabolic systems Commun. P.D.Es 22 3&4 1997 413 433
13. L. Dung H. Smith Steady states of models of microbial growth and competition with chemotaxis J. Math. Anal. Appl. 229 1999 295 318
14. L. Dung H.L. Smith On a parabolic system modelling microbial competition in an unmixed bio-reactor J. Diff. Eqs. 130 1 1996 59 91
15. L. Dung H.L. Smith P. Waltman Growth in the unstirred chemostat with different diffusion rates Fields Inst. Commun. 21 1999 131 142
16. A. Friedman, Partial Differential Equations, New York, 1969
17. J. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc. Math. Surveys and Monographs, 1988, p. 25
18. D. Henry Some infinite-dimensional Morse–Smale systems defined by parabolic differential equations J. Diff. Eqs. 59 1985 165 205
19. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981
20. M. Herrero J. Velasquez Chemotactic collapse for the Keller–Segel model J. Math. Biol. 35 1996 177 194
21. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, vol. 247, Longman Scientific and Technical, 1991
22. W. Jäger S. Luckhaus On explosion of solutions to a system of partial differential equations modelling chemotaxis Trans. Amer. Math. Soc. 329 2 1992 819 824
23. E.F. Keller L.A. Segel Initiation of slime mold aggregation viewed as an instability J. Theor. Biol. 26 1970 399 415
24. K.H.W. Küfner Invariant regions for quasilinear reaction–diffusion systems and applications to a two population model NoDEA 3 1996 421 444
25. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, AMS Transl. Monographs, 1968, p. 23
26. C.S. Lin W.M. Ni I. Takagi Large amplitude stationary solutions to a chemotaxis system J. Diff. Eqs. 72 1 1988 1 27
27. R. Redlinger Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics J. Diff. Eqs. 118 1995 219 252
28. R. Schaff Stationary solutions of chemotaxis systems Trans. Amer. Math. Soc. 292 1985 531 556
29. H. Smith X.Q. Zhao Dynamics of a periodically pulsed bio-reactor model J. Diff. Eqs. 155 1999 368 404
30. X. Wang Qualitative behavior of solutions of a chemotactic diffusion system: effects of motility and chemotaxis and dynamics SIAM Mathematical Analysis 31 3 2000 535 560