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

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.