Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth. For any positive number $R$ there exists a radially symmetric stationary solution with free boundary $r=R$. The system depends on a positive parameter $\mu$, and for a sequence of values $\mu_2<\mu_3<\cdots$ there also exist branches of symmetric-breaking stationary solutions, parameterized by $\varepsilon$, $\vert\varepsilon\vert$ small, which bifurcate from these values. In particular, for $\mu=\mu(\varepsilon)$ near $\mu_2$ the free boundary has the form $r=R+\varepsilon Y_{2,0}(\theta)+O(\varepsilon^2)$ where $Y_{2,0}$ is the spherical harmonic of mode $(2,0)$. It was recently proved by the authors that the stationary solution is asymptotically stable for any $0<\mu<\mu_*$, but linearly unstable if $\mu>\mu_*$, where $\mu_*=\mu_2$ if $R>\bar R$ and $\mu_*<\mu_2$ if $R<\bar R$; $\bar R\approx 0.62207$. In this paper we prove that for $R>\bar R$ each of the stationary solutions which bifurcates from $\mu=\mu_2$ is linearly stable if $\varepsilon>0$ and linearly unstable if $\varepsilon<0$. We also prove, for $R<\bar R$, that the point $\mu=\mu_*$ is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in $t$.