Large time behavior for the classical wave equation with different regular data and its applications

TL;DR

This paper investigates the large time behavior of solutions to the classical wave equation with various initial data regularities, deriving optimal estimates and applying results to related wave and fluid models.

Contribution

It establishes new large time estimates for the wave equation with different initial data regularities and identifies thresholds for solution stabilization.

Findings

Derived optimal large time estimates for wave solutions with $L^2$ and weighted $L^1$ data.

Discovered thresholds for local and global stabilization of wave solutions.

Applied results to scale-invariant wave equations, $\sigma$-evolution, Moore-Gibson-Thompson, and Euler systems.

Abstract

In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\Delta u=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial data belonging to $L^2$ or with additional weighted $L^1$ integrabilities. Particularly, some thresholds are discovered for the (local or global in time) stabilization of this quantity. We also apply these results to the wave equation with scale-invariant terms, the undamped $\sigma$-evolution equation, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system.