Large-time behavior of solutions to a tumor invasion model of Chaplain–Anderson type with quasi-variational structure

Abstract

We treat 2D and 3D tumor invasion models with quasi-variational structures, which are composed of two PDEs, one ODE and certain constraint conditions. Although the original model was proposed by M. R. A. Chaplain and A. R. A. Anderson in 2003, the difference between their original model and ours is that the constraint conditions for the distributions of tumor cells and the extracellular matrix are imposed in our model, which give a quasi-variational structure. For 2D and 3D tumor invasion models with quasi-variational structures, we show the existence of global-in-time solutions and consider their large-time behaviors. Especially, for the large-time behaviors, we show that there exists at least one global-in-time solution such that it converges to a constant steady state in an appropriate function space as time goes to $\infty$.