Well-Posedness and Asymptotic Stability of a Non-Linear Porous System with a Delay Term
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Authors: H. Makheloufi, N. Mezouar and M. Bahlil
DOI: 10.46793/KgJMat2404.591M
Abstract:
Our interest in this work is to treat a one-dimensional Porous system with a non-linear damping and a delay in the non-linear internal feedback. We prove the global existence and uniqueness of its solution in suitable function spaces by means of the Faedo-Galerkin procedure combined with the energy method under a suitable relation between the weight of the delayed feedback and the weight of the non-delayed feedback. Also, we give an explicit and general decay rate estimate by applying the well-known multiplier method integrated with some properties of convex functions and for two opposites cases with respect to the speeds of wave propagation.
Keywords:
Non-linear Porous system, global existence, delay term, general decay, Faedo-Glaerkin method, multiplier method.
References:
[1] F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 643–699. https://doi.org/10.1007/s00030-007-5033-0
[2] T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Appl. Anal. 10 (2015), 187–202. https://doi.org/10.1080/00036811.2014.1000314
[3] T. A. Apalara, A general decay for a weakly nonlinearly damped porous system, J. Dyn. Control Syst. 25 (2019), 311–322. https://doi.org/10.1007/s10883-018-9407-x
[4] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
[5] A. Benaissa and M. Bahlil, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math. 18(5) (2014), 1411–1437. https://doi.org/10.11650/tjm.18.2014.3586
[6] M. Kirane, B. Said-Houari, and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal. 10(2) (2011), 667–686. https://doi.org/10.3934/cpaa.2011.10.667
[7] H. Khochemane, S. Zitouni and L. Bouzettouta, Stability result for a nonlinear damping Porous-elastic system with delay term, Nonlinear Stud. 27(2) (2020), 487–503. https://doi.org/10.46793/KgJMat2003.443S
[8] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson Wiley, Paris, 1994.
[9] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969.
[10] W. J. Liu and M. M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time varying delay term in the internal feedback, Contin. Mech. Thermodyn. 29 (2017), 731–746. https://doi.org/10.1007/s00161-017-0556-z
[11] M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Sys. 16(2) (2010), 211–226. https://doi.org/10.1007/s10883-010-9090-z
[12] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 41 (2011), 1–20.
[13] R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity, Appl. Math. Lett. 16(4) (2003), 487–491.
[14] C. A. Raposo T. A. Apalara and J. O. Ribeiro, Analyticity to transmission problem with delay in Porous-elasticity, J. Math. Anal. App. 466 (2018), 819–834. https://doi.org/10.1016/J.JMAA.2018.06.017
[15] C. A. Raposo J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18(5) (2005), 535–541. https://doi.org/10.1016/j.aml.2004.03.017
[16] A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations 29 (2003), 1–14.
[17] S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philisophical Magazin 41(245) (1921), 744–746. https://doi.org/10.1080/14786442108636264
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