A Free Boundary Problem Related to Condensed Two-Phase Combustion. Part II. Stability and Bifurcation

A family of differential equations in a Banach space is studied. By using invariant manifold and invariant foliation theories, a complete discussion about the stability of a family of equilibria is given. A new bifurcation scenario is discovered in such a way that a one-parameter family of equilibria bifurcates into pieces of cylindrical type surface with spiral flows. As an application to bifurcations from traveling wave solutions of general one space dimension two-phase nonlinear free boundary problems, we show that the bifurcating cylindrical type surface pieces from the traveling wave solutions connect together in a smooth way. Moreover, the flow on the global connected surface winds around with a periodic speed. The application to bifurcations from traveling wave solutions of general semi-linear parabolic equations is also discussed. Similar results are obtained.