Lyapunov functions and Lp-estimates for a class of reaction-diffusion systems

Colloquium Mathematicum

Volumn 87, Issue 1, 2001, Pages 113-127

  Horstmann, Dirk ^a  

^a UNIVERSITY OF COLOGNE   (Germany)

Author keywords

Asymptotic behaviour of solution; Local existence of solution; Lyapunov functions; Uniform Lp estimates

Indexed keywords

EID: 0005525774     PISSN: 00101354     EISSN: 17306302     Source Type: Journal    
DOI: 10.4064/cm87-1-7     Document Type: Article

References (23)

1. p bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827-868.
2. H. Amann, Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), 13-75.
3. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte Math. 133, Teubner, 1993, 9-126.
4. P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles, III , Colloq. Math. 68 (1995), 229-239.
5. P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.
6. P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behaviour of solutions, Nonlinear Anal. 23 (1994), 1189-1209.
7. P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, II , Colloq. Math. 67 (1994), 287-308.
8. P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I , Colloq. Math. 66 (1994), 319-334.
9. S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), 217-237.
10. H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195 (1998), 77-114.
11. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981.
12. D. Horstmann, Aspekte positiver Chemotaxis, Dissertation, Universität zu Köln, 1999.
13. D. Horstmann, On the existence of radially symmetric blowup-solutions for the so-called KellerSegel model , NoDEA Nonlinear Differential Equations Appl., to appear.
14. D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results, preprint, 1999, submitted.
15. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.
16. J. D. Murray, Modelling the pattern generating mechanism in the formation of stripes on alligators, in: Mathematical Physics, 9th Int. Congr. Swansea/UK 1988, (1989), 208-213.
17. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581-601.
18. T. Nagai, T. Senba and K. Yoshida, Application of the Moser-Trudinger inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), 411-433.
19. V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol. 42 (1973), 63-105.
20. K. Post, A non-linear parabolic system modeling chemotaxis with sensitivity functions, preprint no. 99-12, Humboldt-Universität zu Berlin, Institut für Mathematik, 1999.
21. R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292 (1985), 531-556.
22. G. Wolansky, A critical parabolic estimate and application to nonlocal equations arising in chemotaxis, Appl. Anal. 66 (1997), 291-321.
23. A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon. 45 (1997), 241-265.