Global existence and boundedness in a chemotaxis–haptotaxis system with signal-dependent sensitivity

Abstract

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity{ut=Δu−∇⋅(χ(v)u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rn(n≥3) is a bounded domain with smooth boundary, ξ,μ>0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4], Tao [10], Tao and Winkler [11]); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system.