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Energy structure and asymptotic profile of the linearized system of thermo-elastic equation in lower space dimensions. (English) Zbl 1435.35376
Kato, Keiichi (ed.) et al., Asymptotic analysis for nonlinear dispersive and wave equations. Proceedings of the international conference on asymptotic analysis for nonlinear dispersive and wave equations, Osaka University, Osaka, Japan, September 6–9, 2014. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 81, 101-120 (2019).
Summary: 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 ([the first author, “Asymptotic profile of a solution to thermo-elastic equations in two space dimension”, submitted]). 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.
For the entire collection see [Zbl 1435.35006].
MSC:
35Q74 PDEs in connection with mechanics of deformable solids
74B15 Equations linearized about a deformed state (small deformations superposed on large)
74F05 Thermal effects in solid mechanics
74B10 Linear elasticity with initial stresses
35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
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