Peter W. Bates Michigan State University (USA)
Invariant Manifolds of Spikes
Many singularly perturbed nonlinear elliptic equations have spikelike stationary solutions. These can be found
through various methods, including LyapunovSchmidt schemes that, in the neighborhood of a
proposed spike solution, decompose the operator equation into one that is restricted to a “normal
subspace” and one in a “tangential subspace”. Here, these subspaces correspond to eigenstates of the
operator, linearized at an approximate spike solution, and where “tangential” means “corresponding
to eigenvalues near zero”, and “normal” means “complementary”. In this talk I will describe a
more global decomposition in which the “tangential subspace” is replaced by a finitedimensional
manifold of spikelike states and this manifold is invariant with respect to the corresponding nonlinear
parabolic equation and also normally hyperbolic. The stationary spikelike states lie on this manifold.
