Blow-up of solution for an integro-differential equation with arbitrary positive initial energy

In this paper, we consider the integro-differential equation utt−M(∥∇u∥22)Δu+∫0tg(t−τ)Δu(τ)dτ+ut=f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{tt}-M(\|\nabla u\|^{2}_{2})\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\, d\tau+ u_{t}=f(u)$\end{document}, (x,t)∈Ω×(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x,t)\in\Omega\times(0,T)$\end{document}, with initial and Dirichlet boundary conditions. Under suitable assumptions on the functions g and the initial data, a blow-up result with arbitrary positive initial energy is established.