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 0<γ≤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 2n<γ≤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.