Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay

In this paper, we consider initial-boundary value problem of Euler–Bernoulli viscoelastic equation with a delay term in the internal feedbacks. Namely, we study the following equation utt(x,t)+Δ2u(x,t)-∫0tg(t-s)Δ2u(x,s)ds+μ1ut(x,t)+μ2ut(x,t-τ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s){\rm d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0 $$\end{document}together with some suitable initial data and boundary conditions in Ω×(0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega\times (0,+\infty)}$$\end{document} . For arbitrary real numbers μ1 and μ2, we prove that the above-mentioned model has a unique global solution under suitable assumptions on the relaxation function g. Moreover, under some restrictions on μ1 and μ2, exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.