The Global Existence and Boundedness of Solutions to a Chemotaxis–Haptotaxis Model with Nonlinear Diffusion and Signal Production

Abstract

In this paper, we investigate the following chemotaxis–haptotaxis system (1) with nonlinear diffusion and signal production under homogenous Neumann boundary conditions in a bounded domain with smooth boundary. Under suitable conditions on the data we prove the following: (i) For 0γ≤2n, if α>γ−k+1 and β>1−k, problem (1) admits a classical solution (u,v,w) which is globally bounded. (ii) For 2nγ≤1, if α>γ−k+1e+1 and β>max{(nγ−2)(nγ+2k−2)2n−k+1,(nγ−2)(γ+1e)n−k+1} or α>γ−k+1 and β>max{(nγ−2)(nγ+2k−2)2n−k+1,(nγ−2)(α+k−1)n−k+1}, problem (1) admits a classical solution (u,v,w) which is globally bounded.

Keywords

• boundedness
• chemotaxis–haptotaxis
• nonlinear diffusion
• signal production