Global existence and decay property of the Timoshenko system in thermoelasticity with second sound. (English) Zbl 1254.35217
This paper is concerned with the global existence and asymptotic behavior of smooth solutions to a nonlinear one-dimensional Timoshenko system in thermoelasticity with second sound. As is well-known that the linearized system of the nonlinear Timoshenko system is of regularity-loss type, which creates difficulties when dealing with the nonlinear problem. In fact, the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. In this paper, the authors use an energy method with negative weights to create an artificial damping which allows to control the nonlinearity, and prove that for \(0\leq k\leq [s/2]-2\) with \(s\geq 8\), a unique smooth solution exists globally in time and decays as O\(((1+t)^{-1/4-k/2})\) provided the initial datum in \(H^s\cap L^2\), thus generalizing the known result on initial boundary value problems in a bounded domain to the Cauchy problem.
Reviewer: Song Jiang (Beijing)
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 74D05 | Linear constitutive equations for materials with memory |
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