Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis

The following degenerate parabolic system modelling chemotaxis is considered.(KS)τut=∇·(∇um-u∇v),x∈RN,t>0, τvt=Δv-v+u,x∈RN,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈RN,where τ=0 or 1. We show here that the system of (KS)τ with m>1 has a time global weak solution (u,v) with a uniform bound in time when (u0,v0) is a nonnegative function and in L1∩L∞(RN)×L1∩H1∩W1, ∞(RN),u0m∈H1RN). The decay properties of the solution with small initial data are also discussed.