Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains

The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping uₜₜ+Δ²u-∫₀ᵗg(t-τ )Δ²u(τ )\,dτ +a(t )uₜ=0\,\,\,in\,\,\,Ω× R⁺, where a(t)=M(∫Ω|∇ u(x,t)|²dx ), is considered in bounded or unbounded domains Ω of R^ n. The existence of global solutions and decay rates of the energy are proved.