Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on \mathbb{R}^N

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Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type

December 2013, 18(10): 2537-2568. doi: 10.3934/dcdsb.2013.18.2537

Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$

Sachiko Ishida ^1, , Yusuke Maeda ^1, and Tomomi Yokota ^1,

1.	Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, Japan

Received November 2012 Revised February 2013 Published October 2013

Abstract
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This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.

Keywords: blow-up., Quasilinear non degenerate Keller-Segel systems.

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B3.

Citation: Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537

References:

[1]	H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256.
[2]	H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6.
[3]	T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326* (2007), 1410. 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. 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. 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.
[7]	H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319. 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, (1968).
[9]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (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.
[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.
[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. 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. doi: 10.1016/j.jde.2011.08.019.*

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References:

[1]	H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256.
[2]	H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6.
[3]	T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326* (2007), 1410. 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. 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. 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.
[7]	H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319. 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, (1968).
[9]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (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.
[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.
[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. 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. doi: 10.1016/j.jde.2011.08.019.*

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