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Decay property of the Timoshenko-Cattaneo system. (English) Zbl 1338.35052

Summary: We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal \(L^2\) decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko-Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system, although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko-Cattaneo system.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
74F05 Thermal effects in solid mechanics
35G10 Initial value problems for linear higher-order PDEs
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