Copyright © 2010 Claudio Fernández et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study abstract equations of the form λu′′′(t)+u′′(t)=c2Au(t)+c2μAu′(t)+f(t), 0<λ<μ which is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of (α;β;γ)-regularized families, which is a particular case of (a;k)-regularized families, and characterize maximal regularity in Lp-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.