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,ut=Δum−χ∇·(u∇v)−ξ∇·(u∇ω)+μu(1−u−ω),inΩ×R^(+),vt−Δv+v=u,inΩ×R+,ωt=−vω,inΩ×R+,whereΩ⊂R^(N)is a bounded domain.We first supplement the results of global existence and uniform boundedness of solutions for the case m=2N N+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 coefficientχis large,the equilibrium(1,1,0)become unstable.Then we study the existence of nontrivial stationary solutions via bifurcation techniques withχbeing the bifurcation parameter,and obtain nonhomogeneous patterns.At last,we also investigate the stability of these bifurcation solutions.