Energy decay and blow-up of solutions for a viscoelastic equation with nonlocal nonlinear boundary dissipation

In this paper, we consider a nonlinear viscoelastic equation with nonlocal nonlinear boundary damping and two nonlinear source terms (boundary and interior). We obtain the global existence of solution via the potential well method. By introducing suitable Lyapunov functionals, using the multiplier method, and constructing convex differential inequality, we establish an explicit and general decay rate result. This improves previous decay results concerning the problem. We also get a finite time blow-up result of solution with positive initial energy under suitable conditions on the initial data and positive memory function. This is the first result for blow-up of the problem with nonlocal nonlinear boundary damping.