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

This paper deals with the coupled chemotaxis-haptotaxis model [GRAPHICS] 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, ohm subset of R-2 is a bounded domain with smooth boundary, and chi, xi and eta are positive parameters. As compared to previous mathematical studies, the novelty here consists of allowing for positive values of eta, reflecting processes of self-remodeling of the extracellular matrix. Under zero-flux boundary conditions, it is shown that for any sufficiently smooth initial data (u(0), w(0)) 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 L log L(ohm). This serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions. (C) 2014 Elsevier Inc. All rights reserved.