We prove the $L^p$-$L^q$-time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the $L^p$-$L^q$-time decay estimates.

Some critical remarks are made about the theory of Linear Elasticity, questioning on its validity in general. An alternative linear approximation of the exact theory is proposed.

It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain ${\Omega }_h$ is localized either at the whole lateral surface ${\Gamma }_h$ of the domain, or at a point of ${\Gamma }_h$, while the eigenfunction decays exponentially inside ${\Omega }_h$. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.