Bifurcation from stability to instability for a free boundary problem arising in a tumor model

We consider a time-dependent free boundary problem with radially symmetric initial data: sigma(t) - Delta sigma + sigma = 0 if r < R( t), sigma = (sigma) over bar on r = R(t), R-2(t) R(t) = mu integral(R(t))(0) (sigma - (sigma) over tilde )r(2)dr, and sigma(r, 0) = sigma(0)(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) sigma(r, t) and proliferation rate mu(sigma - (sigma) over tilde). If 0 < (sigma) over tilde < (sigma) over bar, then there exists a unique stationary solution (sigma(S)(r), R-S), where R-S depends only on the number (sigma) over tilde/(sigma) over bar. We prove that there exists a number mu(*), such that if mu < mu(*)... then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if mu > mu(*) then the stationary solution is unstable.