$L^{2}$-estimates for the linear elastic waves

Abstract

This paper is concerned with the large time behavior of the solution to the Cauchy problem for the elastic wave equations. In particular, optimal $L^{2}$ estimates of the elastic waves are obtained in the sense that the upper and lower bounds of the $L^{2}$ norms of each component of the solution are proved for large $t$, under the minimum assumptions necessary regarding regularity with respect to initial data. The proof is based on the approximation of the solution by a smooth auxiliary function with suitable parameters.