Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, g^′(t) ≤ − ξ(t) G(g(t)). We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = s^p and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.