Abstract
We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with δ-interaction on a star graph Γ. 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 δ-interaction. This permits to prove virial identity for the H1-solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-δ′ equation on the line.
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