L2-blowup estimates of the wave equation and its application to local energy decay

Abstract

We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].