Heat–Viscoelastic Plate Interaction via Bending Moment and Shear Forces Operators: Analyticity, Spectral Analysis, Exponential Decay

Abstract

We consider a heat–plate interaction model where the 2-dimensional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each component evolves through differential operators. In this paper, we apply “high” boundary interface conditions, which involve the two classical boundary operators of a physical plate: the bending moment operator \(B_1\) and the shear forces operator \(B_2\). We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point \(\lambda =-\,1\) in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.