Global Solvability, Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

In this paper, we deal with the following chemotaxis-haptotaxis system modeling cancer invasion with nonlinear diffusion {u_t = Δu^m - χ∇· (u∇ v) - ξ∇· (u∇ω ) + μ u(1 - u - ω ), in Ω×ℝ^ + ,v_t - Δ v + v = u, in Ω×ℝ^ + ,ω _t = - vω , in Ω×ℝ^ + ,. where Ω ⊂ ℝ N is a bounded domain. We first supplement the results of global existence and uniform boundedness of solutions for the case m = 2NN + 2 . Then for any m > 0 and any spatial dimension, we consider the stability of equilibrium, and find that the chemotaxis has a destabilizing effect, that is for the ODEs, or the diffusion-ODE system without chemotaxis, the solutions tend to a linearly stable uniform steady state (1, 1, 0). When the chemotactic coefficient χ is large, the equilibrium (1, 1, 0) become unstable. Then we study the existence of nontrivial stationary solutions via bifurcation techniques with χ being the bifurcation parameter, and obtain nonhomogeneous patterns. At last, we also investigate the stability of these bifurcation solutions.