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

Abstract

In this paper, we deal with the following chemotaxis-haptotaxis system modeling cancer invasion with nonlinear diffusion

$$\left\{ {\matrix{{{u_t} = \Delta {u^m} - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla \omega ) + \mu u(1 - u - \omega ),\,\,\,\,\,\,{\rm{in}}\,\,\Omega \times {\mathbb{R}^ + },} \hfill \cr {{v_t} - \Delta v + v = u,\,\,\,\,\,\,{\rm{in}}\,\,\Omega \times {\mathbb{R}^ + },} \hfill \cr {{\omega _t} = - v\omega ,\,\,\,\,\,\,{\rm{in}}\,\,\Omega \times {\mathbb{R}^ + },} \hfill \cr } } \right.$$

where Ω ⊂ ℝ^N is a bounded domain. We first supplement the results of global existence and uniform boundedness of solutions for the case $m = {{2N} \over {N + 2}}$. 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 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.