Decay rates of interactive hyperbolic-parabolic PDE models with thermal effects on the interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic- like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A. specific prototype consis ts of a wave equation defined on a three-dimensional bounded domain Omega c oupled with a thermoelastic plate equation defined on Gamma(0)-a flat surfa ce of the boundary partial derivative Omega. Thus, the coupling between the wave and the plate takes place on the interface To. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniforml y stable, the question becomes: what is the "minimal amount" of dissipation necessary to obtain uniform decay rates for the energy of the overall syst em? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to To, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity i s accounted for in the plate model; (ii) the plate model does not account f or any type of mechanical damping, including the structural damping most of ten considered in the literature; (iii) there is no mechanical damping plac ed on the interface To; (iv) the boundary damping is nonlinear without a pr escribed growth rate at the origin; (v) the undamped portions of the bounda ry a partial derivative Omega are subject to Neumann (rather than Dirichlet ) boundary conditions, which is a recognized difficulty in the context of s tabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propag ated onto the hyperbolic component of the structure. This is achieved by us ing a recently developed sharp theory of boundary traces corresponding to w ave and plate equations, along with the analytic estimates recently establi shed for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analy ticity of the thermoelastic plate component allow one to establish appropri ate inverse/recovery type estimates which are critical for uniform stabiliz ation. Our main result provides "optimal" uniform decay rates for the energ y function corresponding to the full structure. These rates are described b y a suitable nonlinear ordinary differential equation, whose coefficients d epend on the growth of the nonlinear dissipation at the origin.