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Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
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Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
1. | Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, Japan |
References:
[1] |
H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250.
doi: 10.1007/BF01215256. |
[2] |
H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxvi+335 pp.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis. J. Math. Anal. Appl., 326 (2007), 1410-1426.
doi: 10.1016/j.jmaa.2006.03.080. |
[4] |
M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[5] |
S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440.
doi: 10.1016/j.jde.2011.02.012. |
[6] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[7] |
H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains, Math. Ann., 312 (1998), 319-340.
doi: 10.1007/s002080050224. |
[8] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 xi+648 pp. |
[9] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. |
[10] |
Y. Sugiyama, Global existence in the sub-critical cases and finite time blow-up in the super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. |
[11] |
Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. |
[12] |
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), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[13] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
show all references
References:
[1] |
H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250.
doi: 10.1007/BF01215256. |
[2] |
H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxvi+335 pp.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis. J. Math. Anal. Appl., 326 (2007), 1410-1426.
doi: 10.1016/j.jmaa.2006.03.080. |
[4] |
M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[5] |
S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440.
doi: 10.1016/j.jde.2011.02.012. |
[6] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[7] |
H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains, Math. Ann., 312 (1998), 319-340.
doi: 10.1007/s002080050224. |
[8] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 xi+648 pp. |
[9] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. |
[10] |
Y. Sugiyama, Global existence in the sub-critical cases and finite time blow-up in the super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. |
[11] |
Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. |
[12] |
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), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[13] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
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