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

Abstract

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 = M n ( R ) there exist symmetry-breaking bifurcation branches of solutions with free boundary r = R + ε Y n , 0 ( θ ) + O ( ε 2 ) ( n even ⩾2) for small | ε | , where Y n , 0 is the spherical harmonic of mode ( n , 0 ) . Furthermore, the smallest M n ( R ) , say M n ∗ ( R ) , is such that n ∗ = n ∗ ( R ) → ∞ as R → ∞ . In this paper we prove that the radially symmetric stationary solution with R = R S is linearly stable if μ / γ < N ∗ ( R S , γ ) and linearly unstable if μ / γ > N ∗ ( R S , γ ) , where N ∗ ( R S , γ ) ⩽ M n ∗ ( R S ) , 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.