Abstract
We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.
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Kunze, M., Schneider, G. Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum . Z. angew. Math. Phys. 55, 383–399 (2004). https://doi.org/10.1007/s00033-004-1099-2
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DOI: https://doi.org/10.1007/s00033-004-1099-2