$L^{2}$-blowup estimates of the wave equation and its application to local energy decay

TL;DR

This paper derives sharp L^{2}-blowup estimates for the wave equation in one and two dimensions with weighted initial data and applies these results to improve local energy decay estimates, especially when the initial velocity's zero-th moment is non-zero.

Contribution

It introduces new sharp blowup estimates for the wave equation with weighted initial data and extends local energy decay results to cases with non-vanishing initial velocity moments.

Findings

Sharp infinite time blowup estimates in 1D and 2D

Enhanced local energy decay results for non-zero initial velocity moments

Methodology inspired by previous techniques for wave equations

Abstract

We consider the Cauchy problems in the whole space for the wave equation with a weighted L^{1}-initial data. We first derive sharp infinite time blowup estimates of the L^{2}-norm of solutions in the one and two dimensional cases. Then, we apply it to the local energy decay estimates for n = 2, which is not studied so completely when the 0-th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in the author's previous papers.