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)tau {u(t) = del center dot (del u(m) - u del v), x is an element of R-N, t > 0, tau v(t) = Delta v - v + u, x is an element of R-N, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), x is an element of R-N, where tau = 0 or 1. We show here that the system of (KS)(tau) with m > 1 has a time global weak solution (u, v) with a uniform bound in time when (u(0), v(0)) is a nonnegative function and in L-1 boolean AND L-infinity (R-N) x L-1 boolean AND H-1 boolean AND W-1,W-infinity (R-N), u(0)(m) is an element of H-1 R-N). The decay properties of the solution with small initial data are also discussed. (C) 2005 Published by Elsevier Ltd.