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2007 Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis
Yoshie Sugiyama
Differential Integral Equations 20(2): 133-180 (2007).

Abstract

The following degenerate parabolic system modelling chemotaxis is considered. $$ {\mbox{(KS)}} \qquad\qquad \left\{ \begin{array}{llll} & u_t = \nabla \cdot \Big( \nabla u^m - u \nabla v \Big), & x \in \mathbb R^N, \ 0 <t <T, \nonumber \\ & \tau v_t = \Delta v - v + u, & x \in \mathbb R^N, \ 0 <t <T, \nonumber \\ & u(x,0) = u_0(x), \quad \tau v(x,0) = \tau v_0(x), & x \in \mathbb R^N, \end{array} \right. $$ where $m>1, \tau=0$ or 1, and $N \ge 1$. Our aim in this paper is to prove the existence of a global weak solution of (KS) under some appropriate conditions on $m$ without any restriction on the size of the initial data. Specifically, we show that a solution ($u,v$) of (KS) exists globally in time if either (i) $m \ge 2 $ for large initial data or (ii) $1 < m \le 2-\frac{2}{N}$ for small initial data. In the case of (ii), the decay properties with the optimal rate of the solution ($u,v$) are also discussed.

Citation

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Yoshie Sugiyama. "Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis." Differential Integral Equations 20 (2) 133 - 180, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35241
MathSciNet: MR2294463

Subjects:
Primary: 35K57
Secondary: 35B30 , 35B45 , 35B65 , 35D05 , 35Q80 , 92C17

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 2 • 2007
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