APPLICATION OF WEAK STABILIZATION THEORY FOR DEGENERATE PARABOLIC EQUATIONS IN DIVERGENCE FORM TO A CHEMOTAXIS MODEL FOR TUMOR INVASION

This paper deals with the chemotaxis model for tumor invasion, { u(t) = del . (D(u, w) del u - u del v), x is an element of Omega, t > 0, vt = Delta v + wz, x is an element of Omega, t > 0, wt = -wz, x is an element of Omega, t > 0, zt = Delta z - z + u, x is an element of Omega, t > 0 under the conditions that (D(u, w) del u-u del v).nu = del v.nu =del z.nu = 0 on partial derivative Omega and that (u, v, w, z)|t=0 = (u(0), v(0), w(0), z(0)), where Omega subset of R-N (N = 1, 2, 3) is a bounded domain with smooth boundary partial derivative Omega and nu is the outward normal vector to partial derivative Omega. The function D is supposed to satisfy D(0, sigma 2) = 0 and D(sigma(1), sigma(2)) >= (D) over bar (sigma(1)), where (D) over bar (sigma(1)) = k sigma(m-1)(1) (sigma(1) <= sigma(0)), (D) over bar (sigma(1)) = k sigma(m-1)(0) (sigma(1) >= sigma(0)) for some k > 0, m > 1 and s0 > 0, which means that the first equation is a degenerate parabolic equation. The previous paper [4] with Fujie and Ito proved global existence and boundedness of weak solutions to the same system, however, an imperfect stabilization property was established. The purpose of this paper is to apply weak stabilization theory in [8] to obtain that u(center dot, t) -> (u(0)) over bar weakly* in L-infinity( O) as t -> 8, where (u(0)) over bar := 1/|Omega| integral(Omega) (u0).