Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis

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.