A heat–structure interaction model with (formal) ‘square-root’ damping: analyticity and uniform stability

In Part I, the present paper studies a homogeneous, uncontrolled 2D or 3D heat–structure interaction model, where the structure is modeled by an elastic system with (formally) ‘square-root’ damping, and where the two components are subject to high-level coupled conditions at the interface between the two media. Physically, the model occupies a doughnut-like domain: the heat (fluid) occupies the exterior domain, while the elastic structure occupies an interior subdomain. The novelty over past literature is the (formal) ‘square root’ damping of the structure versus either no damping at all or else Kelvin–Voigt (viscoelastic) damping. It is shown that such homogeneous (uncontrolled) model generates a strongly continuous contraction semigroup on a natural energy space, which moreover is analytic and uniformly stable. Next, the paper provides a characterization of the domain of a fractional power related to the generator. This result is then used to study, in Part II, the corresponding non-homogeneous model subject to control action at the interface between the two media and provide for it an optimal regularity result. The choice of the heat component over the (linearized) Navier–Stokes fluid component is only a preliminary step for initial simplicity. The fluid-model introduces serious conceptual and technical difficulties. How to overcome them has been accomplished in past literature [Avalos, G., Triggiani, R.: AMS Contemp. Math. Fluids Waves 440, p. 15-55 (2007); Triggiani, R.: Mathematical Theory of Evolutionary Fluid-Flow Structure Interaction. p. 53-172. vol. 48 (2018)] and will guide a subsequent publication.