## A Hopf bifurcation of interfaces in a Dirichlet problem.(English)Zbl 0936.35197

Summary: We consider a free boundary problem which is reduced from the propagator-controller system satisfying the Dirichlet boundary condition when a layer parameter is equal to zero. We show that the equilibria $$(v^*(x), s^*)$$ exist and there is a Hopf bifurcation at a critical point $$\tau$$ only for $$s^*\in(\frac 13, 1)$$.

### MSC:

 35R35 Free boundary problems for PDEs 35B32 Bifurcations in context of PDEs 37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems 35K57 Reaction-diffusion equations
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