Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model
Avner Friedman, Bei Hu · Transactions of the American Mathematical Society · 2008
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 R there exists a radially symmetric stationary solution with free boundary r = R r=R . The system depends on a positive parameter μ \mu , and for a sequence of values μ 2 > μ 3 > ⋯ \mu _2>\mu _3>\cdots there also exist branches of symmetric-breaking stationary solutions, parameterized by ε \varepsilon , | ε | |\varepsilon | small, which bifurcate from these values. In particular, for μ = μ ( ε ) \mu =\mu (\varepsilon ) near μ 2 \mu _2 the free boundary has the form r = R + ε Y 2 , 0 ( θ ) + O ( ε 2 ) r=R+\varepsilon Y_{2,0}(\theta )+O(\varepsilon ^2) where Y 2 , 0 Y_{2,0}