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

Article type: Research Article

Authors: Chen, Wenhui^{; *}

Affiliations: Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany. E-mail: wenhui.chen.math@gmail.com

Correspondence: [*] Corresponding author. E-mail: wenhui.chen.math@gmail.com.

Abstract: 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 Lm regularity and Lp−Lq 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 L2−L2 estimates with additional Lm 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.

Keywords: Elastic waves, Kelvin–Voigt damping, decay property, asymptotic profile, weakly coupled system, global existence

DOI: 10.3233/ASY-191548

Journal: Asymptotic Analysis, vol. 117, no. 1-2, pp. 113-140, 2020