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

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−k+1 and β>1−k, problem (1) admits a classical solution (u, v, w) which is globally bounded. (ii) For 2nγ−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.