Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant

Abstract

This paper deals with the coupled chemotaxis–haptotaxis model{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,0=Δv+u−v,x∈Ω,t>0,wt=−vw+ηw(1−w−u),x∈Ω,t>0, which was initially proposed by Chaplain and Lolas (2006) [10] to describe the interactions between cancer cells, the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue (extracellular matrix). Here, Ω⊂R2 is a bounded domain with smooth boundary, and χ, ξ, μ and η are positive parameters. As compared to previous mathematical studies, the novelty here consists of allowing for positive values of η, reflecting processes of self-remodeling of the extracellular matrix.Under zero-flux boundary conditions, it is shown that for any sufficiently smooth initial data (u0,w0) satisfying first-order compatibility conditions, the model admits a unique global smooth solution. A crucial ingredient in the proof is an energy-like inequality which, given T>0, yields boundedness of u(⋅,t) in LlogL(Ω). This serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions.