Semigroup stability of the interaction of an undamped plate and a viscoelastic plate

¹ Stanford University, Stanford, CA 94305, USA
² Department of Mathematics and Statistics University of Minnesota Duluth, MN 55812, USA
³ Department of Mathematics and Statistics Florida International University Modesto Maidique Campus, Miami, FL 33199, USA

^* Corresponding author: zliu@d.umn.edu

Received: 8 May 2024
Accepted: 23 September 2025

Abstract

In this note, we investigate stability issues for two coupled plate equations in a bounded domain. First, we consider the interaction of an undamped Euler–Bernoulli plate equation and a Kirchhoff plate equation with structural damping or Kelvin-Voigt damping. This system is velocity coupled and the coupling involves the Laplace operator. We prove that the underlying semigroup is polynomially stable with a decay rate O(t⁻¹) as t goes to infinity or exponentially stable, respectively. Next, we apply the structural damping or Kelvin-Voigt damping to the Euler–Bernoulli plate while the Kirchhoff plate is undamped. We prove that the underlying semigroup is polynomially stable with a decay rate O(t^−1/2 ) in both cases as t goes to infinity. Finally, we analyze the spectrum of all those systems, and the spectrum asymptotics show that our polynolmial stability results are optimal, and none of the four systems exhibits smoothing effects. Our proofs of the stability results rely on appropriate estimates of the corresponding resolvent operators along the imaginary axis, either by a direct estimate or by a contradiction argument.