A viscoelastic wave equation with delay and variable exponents: existence and nonexistence

This article deals with the existence and nonexistence of solutions for a viscoelastic wave equation with time delay and variable exponents on the damping and on source term. Firstly, we get the existence of weak solutions by combining the Banach contraction mapping principle and the Faedo–Galerkin method under suitable assumptions on the variable exponents m·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\left( \cdot \right) $$\end{document} and p·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\left( \cdot \right) $$\end{document}. For nonincreasing positive function g, we obtain the nonexistence of solutions with negative initial energy in appropriate conditions.