Exponential stability of a mildly dissipative viscoelastic plate equation with variable density

Abstract

This paper is concerned with a viscoelastic Kirchhoff plate featuring variable material density. It is modeled by the equation ϱutt−Δutt+normalΔ2u−M∫normalΩ|∇ufalse|2dxΔu−∫0tg(t−s)normalΔ2u(s)ds=0, defined in a bounded domain of RN, where ϱ = |ut|ρ accounts for a velocity‐dependent material density. It is known that its analogue second‐order wave equation can be exponentially stabilized with the sole dissipation given by the memory term. However, for the plate equation, exponential stability was only shown with an additional strong damping −Δut. Our objective is to show the exponential stability of the present system by exploring only the memory term.