Well-posedness and stability for a viscoelastic wave equation with density and time-varying delay in $\mathbb {R}^n$

Abstract

This paper concerns the well-posedness and energy decay of a linear wave equation with density, infinite memory and time-varying delay in the whole space $\mathbb {R}^n$ $(n\geq 3)$. We consider the weighted spaces $\mathcal {D}^{1,2}(\mathbb {R}^n)$ and $L^2_{\rho }(\mathbb {R}^n)$ introduced by Karachalios and Stavrakakis (1999) to overcome the difficulty that some operators on $\mathbb {R}^n$ are not compact. We prove the global well-posedness of the Cauchy problem by using Faedo-Galerkin approximation and establish the exponential decay of energy when the amplitude of the time delay term is small by using suitable Lyapunov functional.