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

• Published in: Applied mathematics & optimization Vol. 82; no. 2; pp. 755 - 797
• Main Author: Triggiani, Roberto
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2020; Springer Nature B.V
• Subjects: Bending moments; Calculus of Variations and Optimal Control; Optimization; Control; Damping; Decay rate; Differential equations; Interaction models; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operators (mathematics); Perturbation methods; Shear forces; Simulation; Systems Theory; Theoretical; Two dimensional models; Viscoelasticity; Heat–viscoelastic interaction; Analyticity; Exponential decay
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-018-9547-0

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 λ = - 1 in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.