Blow-up and strong instability of standing waves for the NLS-δ equation on a star graph

We study strong instability (by blow-up) of the standing waves for the nonlinear Schrodinger equation with d-interaction on a star graph Gamma. The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines d-interaction. This permits to prove virial identity for the H-1-solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-delta' equation on the line. (C) 2020 Elsevier Ltd. All rights reserved.