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

The linear Euler-Bernoulli viscoelastic equation$u_{t t} +\Delta^2 u-\int_0^t g(t-\tau)\Delta^2 u(\tau)d\tau = 0\quad$ in $\Omega \times (0,\infty)$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.