Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatiallyhomogeneous equilibrium under the assumption that the associatedlinearization possesses continuous spectrum up to the imaginaryaxis, for all values of the bifurcation parameter, and that a realeigenvalue crosses the imaginary axis. For a model we investigatethe nonlinear stability of the trivial solution with respect tospatially localized perturbations, prove the occurrence of apitchfork bifurcation of equilibria and the nonlinear stability ofthe bifurcating equilibria, again with respect to spatiallylocalized perturbations.