Global Strong Solution and Energy Decay for a Second Order Differential Equation with Memory and Weak Damping

Abstract

In this paper, we consider a differential system with memory and weak nonlinear damping. First, we establish the global existence and uniqueness of strong solution by using the Faedo Galerkin method combined with some compactness results due to the Banach–Alaoglu theorem and the argument of the monotonicity of the logarithmic damping term. In the second part, we establish an uniform stability result with decay governed by relaxation function of memory term. The present result shows that weak dissipation produced by logarithmic damping ensures the same decay as in \cite{26}, which emphasizes the significance of the obtained estimate.