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

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.