zbMATH — the first resource for mathematics

Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type. (English) Zbl 1227.35097
The authors consider the Cauchy problem for a Keller-Segel system where the principal part is of the fast diffusion type. They construct global weak solutions with small initial data and show the existence of a blow-up solution in 2D. Decay and extinction are studied as well.

35B44 Blow-up in context of PDEs
35K59 Quasilinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
92C17 Cell movement (chemotaxis, etc.)
35B40 Asymptotic behavior of solutions to PDEs
Cauchy problem
Full Text: DOI
[1] Aronszajn, N.; Smith, K.T., Theory of Bessel potentials. part I, Ann. inst. Fourier (Grenoble), 11, 385-475, (1961) · Zbl 0102.32401
[2] Bénilan, P.; Crandall, M.G., The continuous dependence on φ of solutions of ut−δφ(u)=0, Indiana univ. math. J., 30, 161-177, (1981) · Zbl 0482.35012
[3] Cieślak, T.; Winkler, M., Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21, 1057-1076, (2008) · Zbl 1136.92006
[4] Galaktionov, V.A.; Peletier, L.A.; Vázquez, J.L., Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. math. anal., 31, 1157-1174, (2000) · Zbl 0954.35094
[5] Herrero, M.A.; Pierre, M., The Cauchy problem for \(u_t = \operatorname{\Delta} u^m\) when \(0 < m < 1\), Trans. amer. math. soc., 291, 145-158, (1985) · Zbl 0583.35052
[6] Kozono, H., \(L^1\)-solutions of the Navier-Stokes equations in exterior domains, Math. ann., 312, 319-340, (1998) · Zbl 0920.35108
[7] Kuroda, S.T., Supekutoru-riron. II, Iwanami shoten kiso Sūgaku, vol. 17, (1983), (in Japanese)
[8] Luckhaus, S.; Sugiyama, Y., Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems, M2AN math. model. numer. anal., 40, 597-621, (2006) · Zbl 1113.35028
[9] Luckhaus, S.; Sugiyama, Y., Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases, Indiana univ. math. J., 56, 1279-1298, (2007) · Zbl 1118.35006
[10] Kurokiba, M.; Ogawa, T., Finite time blow-up of the solution for the nonlinear parabolic equation of the drift diffusion type, Differential integral equations, 16, 427-452, (2003) · Zbl 1161.35432
[11] Nakao, M., Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear anal., 10, 299-314, (1986) · Zbl 0595.35058
[12] Stein, E.M., Singular integrals and differentiability properties of functions, Princeton math. ser., vol. 30, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501
[13] Sugiyama, Y.; Kunii, H., Global existence of solutions of the Keller-Segel model with a nonlinear chemotactical sensitivity function, RIMS kokyuroku, 1432, 49-54, (2005)
[14] Sugiyama, Y.; Kunii, H., Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. differential equations, 227, 333-364, (2006) · Zbl 1102.35046
[15] Sugiyama, Y., Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis, Nonlinear anal., 63, 1051-1062, (2005)
[16] Sugiyama, Y., Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential integral equations, 9, 841-876, (2006) · Zbl 1212.35240
[17] Sugiyama, Y., Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential integral equations, 20, 133-180, (2007) · Zbl 1212.35241
[18] Sugiyama, Y., Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. differential equations, 12, 124-144, (2007) · Zbl 1171.35068
[19] Y. Sugiyama, Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type, submitted for publication. · Zbl 1321.35107
[20] Sugiyama, Y., ε-regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions, J. math. anal. appl., 364, 51-70, (2010) · Zbl 1186.35021
[21] Sugiyama, Y., On ε-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems, SIAM J. math. anal., 41, 1664-1692, (2009) · Zbl 1201.35065
[22] Vázquez, J.L., Smoothing and decay estimates for nonlinear diffusion equations. equations of porous medium type, Oxford lecture ser. math. appl., vol. 33, (2006) · Zbl 1113.35004
[23] Yosida, K., Functional analysis, (1968), Springer-Verlag · Zbl 0152.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.