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Porous-elastic plates: Fourier versus type III. (English) Zbl 1477.35264

Summary: In this paper we investigate the time decay of the solutions for a thermoelastic plate with voids in the cases when the heat conduction is modeled by the Fourier law and when it is modeled by the type III theory (with and without the inertial term). In all situations we show that, in general, the strong stability holds. In particular, we show slow decay of solutions for the Fourier case, that is, the solutions do not decay exponentially to zero (in general). However, if the coefficients satisfy a new relationship involving the inertial coefficient (singular case), we characterize the exponential decay of solutions. On the other hand, for the type III theory the situation is very different and we prove that generically the solutions decay to zero exponentially. This is another striking aspect when we compare both theories. This difference is a consequence of the couplings appearing in the type III case which are not present in the case of the Fourier law.

MSC:

35Q79 PDEs in connection with classical thermodynamics and heat transfer
74F05 Thermal effects in solid mechanics
74K20 Plates
74B10 Linear elasticity with initial stresses
80A19 Diffusive and convective heat and mass transfer, heat flow
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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