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

First published in Transactions of the American Mathematical Society in volume 360, issue 10, published by the American Mathematical Society.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(*), where mu(*) = mu(2) if R > (R) over bar and mu(*) 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