Qualitative properties of solutions for nonlinear Schr\"odinger equations with nonlinear boundary conditions on the half-line

TL;DR

This paper investigates the complex dynamics of nonlinear Schrödinger equations on the half-line, focusing on blow-up solutions, stabilization, and critical exponents influenced by boundary and interior nonlinearities.

Contribution

It introduces conditions for finite-time blow-up solutions considering nonlinear boundary and interior sources, advancing understanding of solution behaviors in such models.

Findings

Existence of blow-up solutions with negative initial energy.

A sufficient condition for blow-up based on nonlinear powers.

Discussion of stabilization and critical exponents.

Abstract

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 Schr\"odinger 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.