Time Global Existence and Asymptotic Behavior of Solutions to Degenerate Quasi-linear Parabolic Systems for Chemotaxis-Growth Models

In this talk, the following degenerate parabolic system modelling chemotaxis is considered. $(KS): u_t = \nabla (\nabla u^m - u \nabla v ), x in R^N, t>0, \tau v_t = \Delta v - v + u, x in R^N, t>0, u(x,0) = u_0(x), \tau v(x,0) = \tau v_0(x), x \in \R^N,$ where $m>1, \tau=0 or 1$, and $N \ge 1$. We discuss 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, it is discussed that a solution (u,v) of (KS) exists globally in time either (i) $2 \le m$ 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 of the solution ($u,v$) are also presented.