Global existence and nonexistence analyses for a magnetic fractional pseudo-parabolic equation

TL;DR

This paper investigates the existence, uniqueness, decay, and blowup of solutions for a magnetic fractional pseudo-parabolic equation within Orlicz-Sobolev spaces, providing comprehensive analysis based on initial energy levels.

Contribution

It introduces new results on global existence, decay estimates, and finite-time blowup for solutions, utilizing advanced functional analysis techniques in magnetic fractional Orlicz-Sobolev spaces.

Findings

Global solutions exist for subcritical initial energy.

Solutions decay over time under certain energy conditions.

Finite-time blowup occurs for specific critical energy levels.

Abstract

In this paper, we study the initial-boundary value problem for a pseudo-parabolic equation in magnetic fractional Orlicz-Sobolev spaces. First, by employing the imbedding theorems, the theory of potential wells and the Galerkin method, we prove the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy and supercritical initial energy, respectively. Furthermore, we prove the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy and supercritical initial energy, respectively. Specifically, we need to analyze the properties of $\omega$-limits of solutions for supercritical initial energy. Next, we establish the finite time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively. Finally, we discuss the convergence relationship between the global solutions of the evolution problem and the ground state solutions of the corresponding stationary problem.