  In this paper we consider elastic waves with Kelvin-Voigt damping in 2D. For
the linear problem, applying pointwise estimates of the partial Fourier
transform of solutions in the Fourier space and asymptotic expansions of
eigenvalues and their eigenprojections, we obtain 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 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-Voigt damping in 3D, we apply the Helmholtz
decomposition.
