Energy structure and asymptotic profile of the linearized system of thermo-elastic equation in lower space dimensions

Abstract

We consider the thermo-elastic problem in the homogeneous isentropic material. After introducing the Helmholtz free energy of the thermo-elastic body and deriving a first approximation of the motion of elastic body with thermal effect, we show an $L^p$-type dissipative-dispersive estimate of Nishihara-type for the linearized equations and it shows that the solution is asymptotically decomposed into solutions to a linear heat equation, a solution to a linear wave equation of exponentially decaying and a diffusive wave ([6]). Then the sharp asymptotic behavior of the solutions to linearized thermo-elastic equations is shown for the coupled elastic-thermal system. As a by-product, we also obtain the Nishihara-type $L^p$ decay estimate for damped wave equation as the limiting case.