Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line

In this paper, we study the interaction between a nonlinear focusing Robin type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrödinger equations posed on the infinite half line. We construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in the $H^1$-sense. We obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow-up solutions. In addition to the blow-up property, we also discuss the stabilization property and the critical exponent for this model.

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