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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 $\mathbb{R}^N$

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.
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
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show all references

References:
[1]

Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.  Google Scholar

[2]

Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxvi+335 pp. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

Comm. Partial Differential Equations, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.  Google Scholar

[5]

J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[6]

J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[7]

Math. Ann., 312 (1998), 319-340. doi: 10.1007/s002080050224.  Google Scholar

[8]

(Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 xi+648 pp.  Google Scholar

[9]

Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[10]

Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[11]

Differential Integral Equations, 20 (2007), 133-180.  Google Scholar

[12]

J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[13]

J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

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