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) < ... there also exist branches of symmetric-breaking stationary solutions, parameterized by epsilon, vertical bar epsilon vertical bar small, which bifurcate from these values. In particular, for mu = mu(epsilon) near mu(2) the free boundary has the form r = R + epsilon Y-2,Y-0(theta) + O(epsilon(2)) where Y-2,Y-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 > (R) over bar and mu(*) < mu(2) if R < (R) over bar; (R) over bar approximate to 0.62207. In this paper we prove that for R > R each of the stationary solutions which bifurcates from mu = mu(2) is linearly stable if epsilon > 0 and linearly unstable if epsilon < 0. We also prove, for R < (R) over bar, 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.