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

Sachiko Ishida; Yusuke Maeda; Tomomi Yokota

doi:10.3934/dcdsb.2013.18.2537

Article Contents

2013, Volume 18, Issue 10: 2537-2568. Doi: 10.3934/dcdsb.2013.18.2537

This issue Previous Article Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model Next Article Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type

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

Received: October 31, 2012

Revised: January 31, 2013

Published: November 2013

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Abstract

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:

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

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B33.

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References

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