Abstract
In this paper, we consider the porous-elastic equations mixing Kelvin–Voigt dissipation mechanisms and the thermal effect given by Fourier’s law. We prove that the system lacks the exponential decay property for a particular equality between damping parameters. In that direction, we prove the polynomial decay and the optimal decay rate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Quintanilla, R., Ueda, Y.: Decay structures for the equations of porous elasticity in one-dimensional whole space. J. Dyn. Differ. Equ. 32, 1669–1685 (2020)
Almeida Júnior, D..S., Santos, M..L., Muñoz Rivera, J..E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Z. Angew. Math. Phys. 65, 1233–1249 (2014)
Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems (Chapman & Hall/CRC Research Notes in Mathematics Series). Chapman and Hall/CRC, London (1999)
Pazy, A.: Decay structures for the equations of porous elasticity in one-dimensional whole space. In: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc. 284, 847–847 (1984)
Gearhart, L.: Spectral theory for contraction semigroups on Hilbert space. Trans. Am. Math. Soc. 236, 385–385 (1978)
Malacarne, A., Muñoz Rivera, J.E.: Lack of exponential stability to Timoshenko system with viscoelastic Kelvin-Voigt type. Z. Angew. Math. Phys. 67, 67 (2016)
Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Mathematische Annalen 347, 455–478 (2009)
Funding
A. J. A. Ramos thanks the CNPq for financial support through Grant 310729/2019-0. D. S. Almeida Júnior thanks the CNPq for financial support through the project “Impact of the second spectrum of frequency on the stabilization of partially dissipative Timoshenko type systems” by Grant 314273/2020-4. M. M. Freitas thanks the CNPq for financial support through Grant 313081/2021-2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ramos, A.J.A., Almeida Júnior, D.S. & Freitas, M.M. About polynomial stability for the porous-elastic system with Fourier’s law. Z. Angew. Math. Phys. 73, 57 (2022). https://doi.org/10.1007/s00033-022-01678-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-022-01678-z