Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction

Abstract

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide a proof by employing a Kato's method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

arXiv comment22 pages, submitted

References in corpus (6)

• A Tunable Two-impurity Kondo system in an atomic point contact
• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians
• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
• Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
• Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting