Finite time extinction by nonlinear damping for Schrodinger equation

TL;DR

This paper studies the nonlinear damping effects on the Schrödinger equation on compact manifolds, demonstrating finite time extinction of solutions in one dimension and under certain regularity conditions in higher dimensions.

Contribution

It establishes finite time extinction for solutions of the nonlinear damped Schrödinger equation on compact manifolds, including the one-dimensional case and higher dimensions with additional regularity.

Findings

Solutions become zero in finite time in 1D.

Finite time extinction occurs in higher dimensions with regular initial data.

Constructs unique weak solutions for all positive times.

Abstract

We consider the Schrodinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the same result under the assumption of extra regularity on the initial datum.