Boundedness and weak stabilization in a degenerate chemotaxis model arising from tumor invasion

Abstract

This paper is concerned with the degenerate system{ut=∇⋅(f(u,w)∇u−g(u)∇v),x∈Ω,t>0,vt=Δv+wz,x∈Ω,t>0,wt=−wz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0 in a bounded domain Ω⊂RN (N≥2) under the no-flux boundary condition for u and the homogeneous Neumann boundary condition for v,z with non-negative initial data u0,v0,w0,z0. Here, the diffusivity f and the sensitivity g are assumed to fulfill f(u,w)≥um−1 (m>1), 0≤g(u)≤uα(α∈R). It is shown that if α+1<m+4N(N≥2) or α+1=m+4N(N≥3) with small mass of u0, then the system possesses a global bounded weak solution which converges to the constant equilibrium in the weak⁎ topology in L∞(Ω) as t→∞.