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/γ < N∗(RS,γ ) and linearly unstable if μ/γ > N∗(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. © 2006 Elsevier Inc. All rights reserved.