Exponential Stabilization of a Semi Linear Third Order in Time Equation with Memory

We are concerned with a third order in time equation in the presence of viscoelastic effects given by the memory term and with a semi linear source term, posed on a bounded domain Omega subset of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}<^>3 $$\end{document}. Considering three different types of memory in the past history framework, we prove the well-posedness of its solutions as well as the exponential stability of the energy functional. Relaxing some hypotheses on the memory kernel, we improve and extend the results established in the existing literature.