Global existence results for semi-linear structurally damped wave equations with nonlinear convection

TL;DR

This paper establishes the global existence and decay properties of solutions for semi-linear wave equations with structural damping and supercritical nonlinearities, emphasizing the importance of solution space selection due to diffusion and regularity loss.

Contribution

It provides new results on global solutions for semi-linear structurally damped wave equations with supercritical nonlinearities, addressing challenges posed by diffusion and regularity issues.

Findings

Proved global existence of solutions for small initial data.

Established decay rates for solutions over time.

Analyzed the impact of nonlinearities on solution behavior.

Abstract

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain global solutions with sharp decay properties in time. Our main purpose of this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.