Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation

The linear Euler-Bernoulli viscoelastic equationut t +Δ² u-∫₀ᵗ g(t-τ)Δ² u(τ)dτ = 0 in Ω × (0,∞)subject to nonlinear boundary conditions is considered. We prove existence and uniformdecay rates of the energy by assuming a nonlinear and nonlocal feedback acting on theboundary and provided that the kernel of the memory decays exponentially.