Global solutions in a quasilinear parabolic-parabolic chemotaxis system with decaying diffusivity and consumption of a chemoattractant

In this study, we consider the quasilinear parabolic-parabolic chemotaxis model:{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, subject to homogeneous Neumann boundary conditions, where Ω is a convex bounded domain of Rn(n≥2) with smooth boundary. The diffusivity sensitivity D(s) and chemotactic sensitivity S(s) satisfy K1e−β−s≤D(s)≤K2e−β+s and S(s)/D(s)≤K3eαs for s≥0 with constants Ki>0(i=1,2,3), β−≥β+>0, and α∈[0,β+/(n+1)). If the initial data are u0∈C0(Ω¯) and v0∈W1,∞(Ω), then the classical solutions to this model are globally bounded.