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 ≤ 2 n , if α > γ − k + 1 and β > 1 − k , problem (1) admits a classical solution ( u , v , w ) which is globally bounded. (ii) For 2 n < γ ≤ 1 , if α > γ − k + 1 e + 1 and β > max { ( n γ − 2 ) ( n γ + 2 k − 2 ) 2 n − k + 1 , ( n γ − 2 ) ( γ + 1 e ) n − k + 1 } or α > γ − k + 1 and β > max { ( n γ − 2 ) ( n γ + 2 k − 2 ) 2 n − k + 1 , ( n γ − 2 ) ( α + k − 1 ) n − k + 1 } , problem (1) admits a classical solution ( u , v , w ) which is globally bounded.