Research on p order nonlinear half wave Schrödinger equations

Abstract

The objective of our research is to scrutinize the presence and singularity of the comprehensive potent solution for the half-wave Schrödinger equation characterized by  order. Take into account the nonlinear half-wave Schrödinger equations represented by  order: . We apply the Brezis-Gallouet style inequality to attain a logarithmic form of regulation. These rationales will be pivotal in validating our primary theorem within Global Well-Posedness. When discussing global Solutions, we infer the solutions for Schrödinger equations within the  realm when . Specifically, when , we leverage the Strichartz estimations alongside inequalities following the Brezis-Gallouët pattern to ascertain the well-posedness of Schrödinger equations within the  framework. To elucidate the global well-posedness of Schrödinger equations in the energy subspace , we undertake a traditional tactic to craft the frail solution within the  realm, succeeded by the application of a rationale found in existing academic literature to verify the singularity of the frail solution. The coherent progression of the frail solution is derived from the sustained conservation of both mass and energy elements. An analogous strategy is likewise harnessed to affirm the global well-posedness of Schrödinger equations in the  sphere when .