Large Time Behavior in a Chemotaxis Model with Nonlinear General Diffusion for Tumor Invasion

Abstract

This paper deals with a chemotaxis system modeling tumor invasion. In the previous papers [7, 8], the case of linear diffusion was studied via the Duhamel formula using the heat semigroup, whereas this method cannot be applied to the case of nonlinear diffusion. The subject of this paper is to develop an approach to the system with some variants of nonlinear diffusion depending on unknown functions in the two cases of non-degenerate and degenerate diffusions. It is shown that a solution of the above system exists globally in time and remains bounded; moreover, under some condition, the solution approaches to the spatially homogeneous equilibrium as time goes to infinity in a certain sense.