This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type Ⅲ in a bounded domain of Rn(n=1; 2; 3) and in the wholespace Rn. For the rst case, we introduce a decoupled system that allows to reduce theproblem of the asymptotic behavior for the original system to a suitable observabilityinequality for the Lame system. In this way most of the existing results for the classicalsystem of thermoelasticity are shown to hold for this system too. In particular, we showthat: (1) For most domains the energy of the system does not decay uniformly; (2) Undersuitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two spacedimensions, the energy of smooth solutions decays polynomially. For the problem in thewhole space Rn, rst, based on Fourier analysis and Lyapunov's second method, we showthat the energy of longitudinal and thermal waves of the system decays as that of theclassical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decayrate of the high frequency longitudinal and thermal waves of the system.

By means of a Carleman estimate, we obtain the global uniqueness and stability of the solution of an inverse problem of determining a wave source within the H−1-space.

It is well known that the heat equation ut−Δu=fχωin (0, T)×Ω with homogeneousDirichlet boundary conditions is null exactly controllable for any T>0 and any open nonemptysubset ω of Ω. In this note we show that this property may be obtained as a singular limit ofthe exact controllability properties of singularly perturbed damped wave equations with a changingcontroller.

In this paper, we obtain the exact controllability for a class of semilinear plate equations under a superlinear growth assumption on the nonlinearity. For this purpose, we need an explicit observability estimate for the linear plate equation with a potential, which is derived by means of Carleman estimates together with the usual energy estimate.

In this paper, by means of Carleman estimates and the usual energy estimate, we obtain directly two observability inequalities for the linear wave equation with time-variantnonsmooth lower order terms. We do not need any unique continuation property of the linearequation a priori, since this is actually one of the by-products of our analysis. Furthermore, the constantin the observability inequality is estimated by an explicit function of the norm of the involvedcoefficients in the equation. Also, we apply our observability estimates to exact controllability forwave equations.