Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions.

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate λ, which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider "flat stationary solution" in this paper. We first show that there exists a unique flat stationary solution for any λ>0. Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points λ=λk(ρ*) (k≥1). [ABSTRACT FROM AUTHOR]