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. $(KS):u_t=\mathrm{\nabla }(\mathrm{\nabla }{u}^m-u\mathrm{\nabla }v),xin{R}^N,t>0,\tau v_t=\mathrm{\Delta }v-v+u,xin{R}^N,t>0,u(x,0)=u_0(x),\tau v(x,0)=\tau v_0(x),x\in {\text{R}}^N,$ where $m>1,\tau =0or1$, 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.