Blow-up results for the semilinear Schr\"{o}dinger equations with forcing and gradient terms: the critical cases

TL;DR

This paper investigates the critical cases of nonlinear Schrödinger equations with source and gradient terms, proving nonexistence of global solutions and addressing open questions in the field.

Contribution

It introduces a modified test function method tailored to critical NLS problems, establishing nonexistence results in these cases.

Findings

Critical cases lack global weak solutions.

Modified test function method effectively proves nonexistence.

Addresses open questions from prior research.

Abstract

The paper is devoted to the study of critical cases of the nonlinear Schr\"{o}dinger (NLS) equation with source and gradient terms, subsequently providing answers to some open questions posed by Alotaibi et al in [Z. Angew. Math. Phys., 73 (2022), 1-17]. The main results state that in critical cases, the problem under consideration does not have any global-in-time weak solutions. The proof approach is based on modified method of test functions specifically adapted to the nature of considered problems.