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

Abstract

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 $$u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s){\rm d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0 $$ together with some suitable initial data and boundary conditions in $${\Omega\times (0,+\infty)}$$ . For arbitrary real numbers μ 1 and μ 2, 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 μ 1 and μ 2, exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.