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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

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.  Google Scholar

[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.  Google Scholar

[3]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.   Google Scholar

[7]

H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319.  doi: 10.1007/s002080050224.  Google Scholar

[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).   Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar

[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.  Google Scholar

[3]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.   Google Scholar

[7]

H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319.  doi: 10.1007/s002080050224.  Google Scholar

[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).   Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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