Existence and large time behavior for a Keller-Segel model with gradient dependent chemotactic sensitivity

The purpose of this paper is to study the chemotaxis growth system{u(t)=delta u-& nabla;.(u|& nabla;v|(p-2)& nabla;v)+au-bu alpha, ,x is an element of omega,t > 0vt=delta v-v+w,w x is an element of omega,t > 0t=delta w-w+u,x is an element of omega,t > 0, in a smooth bounded domain omega subset of R-n, n >= 2 with nonnegative initial data and homogeneous boundary conditions of Neumann type for u,v and w. We will show that the problem admits a global weak solution when p is an element of(1,n alpha+2n-6 alpha+4/2n-6 alpha+4), 3 alpha-2 <= n <= 4 alpha-2, and when p > 1, n < 3 alpha-2. What is more, under appropriate conditions, this global solution with nonnegative initial data (u0,v0,w0) eventually becomes a classical solution of the system and satisfies u ->(a+/b)(1/alpha-1),v ->(a+/b)(1/alpha-1),w ->(a(+)/b)(1/alpha-1) in L-infinity(omega),