Treffer: Bifurcation analysis of a free boundary problem modelling tumour growth under the action of inhibitors

In this paper we investigate non-radial stationary solutions of a free boundary problem modelling tumour growth under the action of inhibitors. The model consists of two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the velocity of tumour cells and internal pressure. The ratio μ/γ of the proliferation rate μ and the cell-to-cell adhesiveness γ plays the role of the bifurcation parameter. We prove that in certain situations there exists a positive sequence {(μ/γ)n}n≥n* such that for each (μ/γ)n (n even ≥ n*) there exist non-radial stationary solutions bifurcating from the radial stationary solution, while in the other situations there exists at most a finite number of bifurcation points. This is a remarkable difference from the corresponding inhibitor-free model where there always exist infinitely many branches of non-radial stationary bifurcation solutions. Our analysis also indicates that inhibitor supply may lower the ability of tumour invasion, and even make the tumour unaggressive and stable.