On e-regularity theorem and asymptotic behaviors of solutions for keller-se gel systems

SIAM Journal on Mathematical Analysis

Volumn 41, Issue 4, 2009, Pages 1664-1692

  Sugiyama, Yoshie ^a  

^a TSUDA COLLEGE   (Japan)

Author keywords

Blow up; E Regularity; Extension criteria; Hausdorff dimension; Keller segel system; Partial regularity

Indexed keywords

ASYMPTOTIC BEHAVIORS; ASYMPTOTICS; BLOW-UP; GEL SYSTEMS; HAUSDORFF DIMENSION; PARABOLIC SYSTEM; PARTIAL REGULARITY; POROUS MEDIUM; WEAK SOLUTION;

ASYMPTOTIC ANALYSIS; MEASUREMENT THEORY; POROUS MATERIALS;

FRACTALS;

EID: 73349139844     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/080721078     Document Type: Article

References (29)

1. N. ARONSZAJN AND K.T. SMITH, Theory of Bessel potentials. Part I, Ann. Inst. Fourier (Grenoble), 11 (1961), pp. 385-475.
2. L.A. CAFFARELLI AND A. FRIEDMAN, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc., 252 (1979), pp. 99 -113.
3. L.A. CAFFARELLI, R. KOHN, AND L. NIRENbERg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), pp. 771-831.
4. Y. CHEN AND M. STRUWE, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z., 201 (1989), pp. 83-103.
5. S. CHILDRESS AND J.K. PERCUS, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), pp. 217-237.\f1
6. E. DLBENEDETTO, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
7. 1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), pp. 109-124.
8. p and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations , 62 (1986), pp. 186-212.
9. W. JAGER AND S. LUCKHAUS, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), pp. 19-824.
10. M.A. HERRERO AND J.J.L. VELAZQUEZ, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), pp. 583-623.
11. D. HORSTMANN, From 1970 until present: The Keller-Seg el model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math-Verein., 105 (2003), pp. 103-165.
12. E.F. KELLER AND L.A. SEGEL, Initiation of slime mold ag gregation viewed as an instability, J. Theoret. Biol., 26 (1970), pp. 399-415.
13. H. KOZONO AND Y. SUGIYAMA, Strong solutions to the Keller-Segel system with the weak Ln/2 initial data and its application to the blow-up rate, Math. Nachr., to appear, DOI 10.1002/mana.200610835.
14. S.T. KURODA, Spectral theory. II, Iwanami Lect. Fundam. Math., Iwanami Shoten, Tokyo, 17, 1983, (in Japanese).
15. J.L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications I, II, Springer-Verlag, Berlin-Heidelberg-New York, 19 72.
16. M. MIMURA AND T. NAGAI, Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics, SIAM J. Appl. Math., 43(1983),pp.449-464.
17. V. NANJUNDIAH, Chemotaxis signal relaying and aggregation morphology, Theoret. Biol., 42(1973),pp.63-105.
18. T. NAGAI, T. SENBA, AND T. SUZUKI, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30(2000),pp.463-497.
19. R. SCHOEN AND K. UHLENBECK, Regularity of minimizing harmonic maps into the sphere, Invent. Math., 78(1984),pp.89-100.
20. Y. SUGIYAMA, Global existence and decay properties of solutions for some degenerate quasilin-ear parabolic systems modelling chemotaxis, Nonlinea r Anal., 63(2005),pp.1051-1062.
21. Y. SUGIYAMA, Global existence in the sub-critical cases and finite time blow-up in the supercritical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19(2006),pp.841-876.
22. Y. SUGIYAMA AND H. KUNII, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227(2006),pp.333-364.
23. Y. SUGIYAMA, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20(2007),pp.133-180.
24. Y. SUGIYAMA, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12(2007),pp.124-144.
25. Y. SUGIYAMA, Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type, submitted.
26. Y. SUGIYAMA, e-regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions, submitted.
27. E.M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, NJ, 1970.
28. M. STRUWE, The existence of surfaces of constant mean curvature with free boundaries, Acta Math., 160(1988),pp.19-64.
29. G. TALENTI, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 4(1976), pp.353-372.