Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum (Q701753)

The authors investigate the scalar diffusion equatioins of the type \(\partial_t U=LU+ N(U)\), where \(LU=\partial_x^2 U+c\partial_x U+ \alpha r(x) U\) \((x\in \mathbb R^1)\) and \(N(U)=\partial_x U^3 \) with given parameters \(c, \alpha\) and scalar function \(r\). They 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, the nonlinear stability of the trivial solution with respect to spatially localized perturbations, is studied. They prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, with respect to spatially localized perturbations.