Global well-posedness and long-time behavior in a tumor invasion model with cross-diffusion

This paper is concerned with a cross-diffusion tumor invasion model with double-taxis effect. We first investigate the global existence of classical solutions of this model in two-dimensional space. The essential difficulty lies in the second-level taxis effect of immune cells on tumor cells, where chemotactic factor (tumor cells) exhibit their own taxis behavior, the double-taxis effect makes us have to use more detailed analysis and calculation, and some new estimation techniques are used. Subsequently, we also investigate the stability of some equilibria. For small proliferation coefficient, we prove the global asymptotic stability or local asymptotic stability of a semitrivial equilibrium point. While, for the other equilibria, the stability analysis is complicated even for some special cases, and both the double chemotactic coefficients will affect the stability of the solution.