Non uniform decay of the total energy of the dissipative wave equation

Abstract

Kawashita, Nakazawa, and Soga [3] give a necessary condition for the uniform energy decay of the dissipative wave equation whose principal term has constant coefficients. In their proof, they construct asymptotic solutions for a suitable family of the Cauchy data. In this paper, instead of the asymptotic solutions, we consider the semiclassical measure associated with the family and extend this result to the variable coefficient case. Moreover we give some lower bound estimate for the energy decay.