Arbitrary Decay of Energy for a Viscoelastic Problem with Balakrishnan-Taylor Damping

Abstract

In this paper we consider a viscoelastic problem withBalakrishnan-Taylor damping\[ u_{tt} - \left(a + b \left\|\nabla u \right\|^2 + \sigma(\nabla u, \nabla u_t) \right) \Delta u + \int^t_{0} g(t-s) \Delta u(s) \, ds = 0\]with Dirichlet boundary condition. We establish a decay result of the energy ofsolutions for the problem without imposing the usual relation between therelaxation function $g$ and its derivative. This result generalizes earlierones to an arbitrary rate of decay, which is not necessarily of exponential orpolynomial decay.