Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller-Segel systems

We consider the initial-boundary value problem for the degenerate parabolic equation u(t) = del . f((u)del u + g(u, x, t)), x is an element of Omega, t > 0 in a smooth bounded domain Omega subset of R-N (N is an element of N) under the no-flux boundary condition with a non-negative initial data u(0) is an element of L-infinity(Omega). Here f is a non-negative function belonging to C([0, infinity)) boolean AND C-2 ((0, infinity)), and g is a vector-valued function on [0, infinity) x Omega x (0, infinity). It is known that this problem has a global-in-time weak solution by the well-known parabolic theory. This paper shows the stabilization in this problem; in detail, the problem admits a global weak solution which fulfills u(t) -> 1/vertical bar Omega vertical bar integral(Omega) u(0) weakly* in L-infinity (Omega) as t -> infinity.