A general method for proving sharp energy decay rates for memory-dissipative evolution equations

Fatiha Alabau-Boussouira; Piermarco Cannarsa

doi:10.1016/j.crma.2009.05.011

Partial Differential Equations/Optimal Control

A general method for proving sharp energy decay rates for memory-dissipative evolution equations
[Une méthode générale pour obtenir des taux de décroissance de l'énergie des équations d'évolution avec dissipation-mémoire]

Présenté par : Philippe G. Ciarlet

Fatiha Alabau-Boussouira ¹ ; Piermarco Cannarsa ²

¹ L.M.A.M. CNRS-UMR 7122 et INRIA Équipe-projet CORIDA, université Paul-Verlaine-Metz, Ile du Saulcy, 57045 Metz cedex 01, France
² Dipartimento di Matematica, Università di Roma Tor Vergata, 00133 Roma, Italy

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 867-872.

Résumé
Abstract

On étudie le problème de la stabilisation des équations de type hyperbolique par un feedback-mémoire distribué. L'objet de cette Note est de montrer qu'il existe une méthode constructive générale qui permet d'obtenir un taux de décroissance de l'énergie en fonction du comportement au voisinage de l'infini du noyau. Cette méthode permet de retrouver de manière naturelle les résultats connus (cas exponentiel, polynômial, …) mais aussi de définir une classe très générale et quasi-optimale de noyaux à laquelle elle s'applique. Elle permet de montrer sous une condition, aussi très générale, que l'énergie des solutions décroit au moins aussi vite que le noyau à l'infini.

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity.

Reçu le : 2008-11-27
Accepté le : 2009-05-15
Publié le : 2009-07-03

DOI : 10.1016/j.crma.2009.05.011

Affiliations des auteurs :

Fatiha Alabau-Boussouira ¹ ; Piermarco Cannarsa ²

¹ L.M.A.M. CNRS-UMR 7122 et INRIA Équipe-projet CORIDA, université Paul-Verlaine-Metz, Ile du Saulcy, 57045 Metz cedex 01, France
² Dipartimento di Matematica, Università di Roma Tor Vergata, 00133 Roma, Italy

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Fatiha Alabau-Boussouira; Piermarco Cannarsa. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 867-872. doi : 10.1016/j.crma.2009.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.011/

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Muhammad I. Mustafa Energy Decay in a Quasilinear System with Finite and Infinite Memories, Mathematical Methods in Engineering, Volume 23 (2019), p. 235 | DOI:10.1007/978-3-319-91065-9_12
Salah Boulaaras; Alaeddin Draifia; Khaled Zennir General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity, Mathematical Methods in the Applied Sciences, Volume 42 (2019) no. 14, p. 4795 | DOI:10.1002/mma.5693
Salah Boulaaras; Alaeddin Draifia; Mohammad Alnegga Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel, Symmetry, Volume 11 (2019) no. 2, p. 226 | DOI:10.3390/sym11020226
Дмитрий Александрович Закора; Dmitry Aleksandrovich Zakora Представление решений одного интегро-дифференциального уравнения и приложения, Итоги науки и техники. Серия «Современная математика и ее приложения. Тематические обзоры», Volume 171 (2019), p. 78 | DOI:10.36535/0233-6723-2019-171-78-93
E. H. Gomes Tavares; M. A. Jorge Silva; V. Narciso On a decay rate for nonlinear extensible viscoelastic beams with history setting, Applicable Analysis, Volume 97 (2018) no. 11, p. 1916 | DOI:10.1080/00036811.2017.1343940
E. H. Gomes Tavares; M. A. Jorge Silva; T. F. Ma Sharp decay rates for a class of nonlinear viscoelastic plate models, Communications in Contemporary Mathematics, Volume 20 (2018) no. 02, p. 1750010 | DOI:10.1142/s0219199717500109
Khaled Zennir; Mouhssin Bayoud; Svetlin Georgiev Decay of Solution for Degenerate Wave Equation of Kirchhoff Type in Viscoelasticity, International Journal of Applied and Computational Mathematics, Volume 4 (2018) no. 1 | DOI:10.1007/s40819-018-0488-8
M.M. Cavalcanti; V.N. Domingos Cavalcanti; M.A. Jorge Silva; A.Y. de Souza Franco Exponential stability for the wave model with localized memory in a past history framework, Journal of Differential Equations, Volume 264 (2018) no. 11, p. 6535 | DOI:10.1016/j.jde.2018.01.044
Muhammad I. Mustafa General decay result for nonlinear viscoelastic equations, Journal of Mathematical Analysis and Applications, Volume 457 (2018) no. 1, p. 134 | DOI:10.1016/j.jmaa.2017.08.019
Muhammad I. Mustafa Optimal decay rates for the viscoelastic wave equation, Mathematical Methods in the Applied Sciences, Volume 41 (2018) no. 1, p. 192 | DOI:10.1002/mma.4604
D. A. Zakora Exponential Stability of a Certain Semigroup and Applications, Mathematical Notes, Volume 103 (2018) no. 5-6, p. 745 | DOI:10.1134/s0001434618050073
Dmitry Aleksandrovich Zakora Экспоненциальная устойчивость одной полугруппы и приложения, Математические заметки, Volume 103 (2018) no. 5, p. 702 | DOI:10.4213/mzm11703
Marcelo M. Cavalcanti; Valéria N. Domingos Cavalcanti; Irena Lasiecka; Claudete M. Webler Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density, Advances in Nonlinear Analysis, Volume 6 (2017) no. 2, p. 121 | DOI:10.1515/anona-2016-0027
Muhammad I. Mustafa Memory-type plate system with nonlinear delay, Advances in Pure and Applied Mathematics, Volume 8 (2017) no. 4 | DOI:10.1515/apam-2016-0111
Muhammad I. Mustafa; Ghassan A. Abusharkh Plate equations with frictional and viscoelastic dampings, Applicable Analysis, Volume 96 (2017) no. 7, p. 1170 | DOI:10.1080/00036811.2016.1178724
Muhammad I. Mustafa Viscoelastic plate equation with boundary feedback, Evolution Equations Control Theory, Volume 6 (2017) no. 2, p. 261 | DOI:10.3934/eect.2017014
Irena Lasiecka Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics, Journal of Evolution Equations, Volume 17 (2017) no. 1, p. 411 | DOI:10.1007/s00028-016-0353-3
Fatiha Alabau-Boussouira; Yannick Privat; Emmanuel Trélat Nonlinear damped partial differential equations and their uniform discretizations, Journal of Functional Analysis, Volume 273 (2017) no. 1, p. 352 | DOI:10.1016/j.jfa.2017.03.010
Muhammad I. Mustafa Energy decay of dissipative plate equations with memory-type boundary conditions, Asymptotic Analysis, Volume 100 (2016) no. 1-2, p. 41 | DOI:10.3233/asy-161385
Valéria Neves Domingos Cavalcanti; Irena Lasiecka; Arthur Henrique Caixeta On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evolution Equations and Control Theory, Volume 5 (2016) no. 4, p. 661 | DOI:10.3934/eect.2016024
Salah Zitouni; Khaled Zennir On the existence and decay of solution for viscoelastic wave equation with nonlinear source in weighted spaces, Rendiconti del Circolo Matematico di Palermo (1952 -) (2016) | DOI:10.1007/s12215-016-0257-7
Irena Lasiecka; Xiaojun Wang Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy, Zeitschrift für angewandte Mathematik und Physik, Volume 67 (2016) no. 2 | DOI:10.1007/s00033-015-0597-8
Muhammad I. Mustafa; Mohammad Kafini Energy decay for viscoelastic plates with distributed delay and source term, Zeitschrift für angewandte Mathematik und Physik, Volume 67 (2016) no. 3 | DOI:10.1007/s00033-016-0641-3
Jum-Ran Kang Asymptotic behavior to a von Kármán equations of memory type with acoustic boundary conditions, Zeitschrift für angewandte Mathematik und Physik, Volume 67 (2016) no. 3 | DOI:10.1007/s00033-016-0639-x
Khaled Zennir General decay of solutions for damped wave equation of Kirchhoff type with density in $R^{n}$ R n, ANNALI DELL'UNIVERSITA' DI FERRARA, Volume 61 (2015) no. 2, p. 381 | DOI:10.1007/s11565-015-0223-x
Muhammad I. Mustafa Energy decay in thermoelasticity with viscoelastic damping of general type, Acta Mathematica Sinica, English Series, Volume 31 (2015) no. 2, p. 331 | DOI:10.1007/s10114-015-2687-0
Jum-Ran Kang General stability for a von Kármán plate system with memory boundary conditions, Boundary Value Problems, Volume 2015 (2015) no. 1 | DOI:10.1186/s13661-015-0431-4
Jum-Ran Kang A general stability for a von Kármán system with memory, Boundary Value Problems, Volume 2015 (2015) no. 1 | DOI:10.1186/s13661-015-0466-6
Muhammad I. Mustafa; Ghassan A. Abusharkh Plate equations with viscoelastic boundary damping, Indagationes Mathematicae, Volume 26 (2015) no. 2, p. 307 | DOI:10.1016/j.indag.2014.09.005
Irena Lasiecka; Xiaojun Wang Moore–Gibson–Thompson equation with memory, part II: General decay of energy, Journal of Differential Equations, Volume 259 (2015) no. 12, p. 7610 | DOI:10.1016/j.jde.2015.08.052
Mauro Fabrizio; Barbara Lazzari; Roberta Nibbi Asymptotic stability in linear viscoelasticity with supplies, Journal of Mathematical Analysis and Applications, Volume 427 (2015) no. 2, p. 629 | DOI:10.1016/j.jmaa.2015.02.061
Marcelo M. Cavalcanti; André D.D. Cavalcanti; Irena Lasiecka; Xiaojun Wang Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Analysis: Real World Applications, Volume 22 (2015), p. 289 | DOI:10.1016/j.nonrwa.2014.09.016
Gang Li; Yun Sun; Wenjun Liu Arbitrary decay of solutions for a singular nonlocal viscoelastic problem with a possible damping term, Applicable Analysis, Volume 93 (2014) no. 6, p. 1150 | DOI:10.1080/00036811.2013.821111
Marcelo M. Cavalcanti; Valéria N. Domingos Cavalcanti; Irena Lasiecka; Flávio A. Falcão Nascimento Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Continuous Dynamical Systems - B, Volume 19 (2014) no. 7, p. 1987 | DOI:10.3934/dcdsb.2014.19.1987
Kun-Peng Jin; Jin Liang; Ti-Jun Xiao Coupled second order evolution equations with fading memory: Optimal energy decay rate, Journal of Differential Equations, Volume 257 (2014) no. 5, p. 1501 | DOI:10.1016/j.jde.2014.05.018
A. Guesmia; S.A. Messaoudi A new approach to the stability of an abstract system in the presence of infinite history, Journal of Mathematical Analysis and Applications, Volume 416 (2014) no. 1, p. 212 | DOI:10.1016/j.jmaa.2014.02.030
Irena Lasiecka; Xiaojun Wang Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Volume 10 (2014), p. 271 | DOI:10.1007/978-3-319-11406-4_14
Ti-Jun Xiao; Jin Liang Coupled second order semilinear evolution equations indirectly damped via memory effects, Journal of Differential Equations, Volume 254 (2013) no. 5, p. 2128 | DOI:10.1016/j.jde.2012.11.019
Irena Lasiecka; Salim A. Messaoudi; Muhammad I. Mustafa Note on intrinsic decay rates for abstract wave equations with memory, Journal of Mathematical Physics, Volume 54 (2013) no. 3 | DOI:10.1063/1.4793988
Nasser–eddine Tatar A New Class of Kernels Leading to an Arbitrary Decay in Viscoelasticity, Mediterranean Journal of Mathematics, Volume 10 (2013) no. 1, p. 213 | DOI:10.1007/s00009-012-0177-5
Salim A. Messaoudi; Muhammad I. Mustafa A general stability result for a quasilinear wave equation with memory, Nonlinear Analysis: Real World Applications, Volume 14 (2013) no. 4, p. 1854 | DOI:10.1016/j.nonrwa.2012.12.002
Salim A. Messaoudi; Muhammad I. Mustafa A general stability result in a memory-type Timoshenko system, Communications on Pure and Applied Analysis, Volume 12 (2012) no. 2, p. 957 | DOI:10.3934/cpaa.2013.12.957
Fatiha Alabau-Boussouira On Some Recent Advances on Stabilization for Hyperbolic Equations, Control of Partial Differential Equations, Volume 2048 (2012), p. 1 | DOI:10.1007/978-3-642-27893-8_1
Nasser-Eddine Tatar; Yongkun Li Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type, Journal of Applied Mathematics, Volume 2012 (2012) no. 1 | DOI:10.1155/2012/936140
Muhammad I. Mustafa; Salim A. Messaoudi General stability result for viscoelastic wave equations, Journal of Mathematical Physics, Volume 53 (2012) no. 5 | DOI:10.1063/1.4711830
Nasser‐eddine Tatar Oscillating kernels and arbitrary decays in viscoelasticity, Mathematische Nachrichten, Volume 285 (2012) no. 8-9, p. 1130 | DOI:10.1002/mana.201000053
Qian Wan; Ti-Jun Xiao Exponential Stability of Two Coupled Second-Order Evolution Equations, Advances in Difference Equations, Volume 2011 (2011), p. 1 | DOI:10.1155/2011/879649
Aissa Guesmia Asymptotic stability of abstract dissipative systems with infinite memory, Journal of Mathematical Analysis and Applications, Volume 382 (2011) no. 2, p. 748 | DOI:10.1016/j.jmaa.2011.04.079

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