Decay properties and asymptotic profiles for elastic waves with Kelvin-Voigt damping in 2D

Wenhui Chen

Published 2018-11-06Version 1

In this paper we consider elastic waves with Kelvin-Voigt damping in 2D. For the linear problem, applying pointwise estimates in the Fourier space and asymptotic expansion of eigenvalues, we establish sharp energy decay estimates with additional $L^m$ regularity and $L^p-L^q$ estimates on the conjugate line. Furthermore, we derive asymptotic profiles of solutions under different assumptions of the initial data. For the semilinear problem, we use the derived $L^2-L^2$ estimates with additional $L^m$ regularity to prove global (in time) existence of small data solutions to the weakly coupled system. Finally, to deal with elastic waves with Kelvin-Vogit damping in 3D, we apply the Helmholtz decomposition.