Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth in fluid-like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation of tumor cells. The proliferation rate μ and the cell-to-cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius r=R. For a sequence μ/γ=Mn(R) there exist symmetry-breaking bifurcation branches of solutions with free boundary r=R+εYn,0(θ)+O(ε2) (n even ⩾2) for small |ε|, where Yn,0 is the spherical harmonic of mode (n,0). Furthermore, the smallest Mn(R), say Mn∗(R), is such that n∗=n∗(R)→∞ as R→∞. In this paper we prove that the radially symmetric stationary solution with R=RS is linearly stable if μ/γN∗(RS,γ), where N∗(RS,γ)⩽Mn∗(RS), and we prove that strict inequality holds if γ is small or if γ is large. The biological implications of these results are discussed at the end of the paper.