[Stabilité exponentielle et polynômiale pour une équation des ondes par un feedback frontière mémoire avec noyau singulier]
On étudie le problème de la stabilisation d'une équation des ondes par un feedback frontière avec mémoire. Le noyau du feedback est supposé singulier. Dans le cas où le feedback est à la fois frictionnel et avec mémoire, on démontre que l'énergie des solutions décroît exponentiellement. Dans le cas où le feedback est seulement de type mémoire, on montre dans cette Note que l'énergie des solutions décroît polynômialement. Le résultat repose sur l'utilisation d'énergies d'ordre plus élevé (cf. [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation ; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020 ; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150 ; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]) la méthode des multiplicateurs et les propriétés d'une large classe de noyaux singuliers (cf. [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309 ; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]). De plus, notre méthode peut être étendue pour traiter des cas de noyaux non singuliers.
This work is concerned with stabilization of a wave equation stabilized by a boundary feedback. When the feedback is both frictional and with memory, we prove exponential stability of the solutions. In case of a boundary feedback which is only of memory type, uniform stability is not expected. We prove in this latter case, that the solutions decay polynomially. The method is new and uses the method of higher order energies (see [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]), the multiplier method and the properties of a large class of singular kernels. Moreover, our method can be extended to include cases of nonsingular kernels (see [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]).
Accepté le :
Publié le :
@article{CRMATH_2009__347_5-6_277_0,
author = {Alabau-Boussouira, Fatiha and Pr\"uss, Jan and Zacher, Rico},
title = {Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels},
journal = {Comptes Rendus. Math\'ematique},
pages = {277--282},
publisher = {Elsevier},
volume = {347},
number = {5-6},
year = {2009},
doi = {10.1016/j.crma.2009.01.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.01.005/}
}
TY - JOUR AU - Alabau-Boussouira, Fatiha AU - Prüss, Jan AU - Zacher, Rico TI - Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels JO - Comptes Rendus. Mathématique PY - 2009 SP - 277 EP - 282 VL - 347 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.01.005/ DO - 10.1016/j.crma.2009.01.005 LA - en ID - CRMATH_2009__347_5-6_277_0 ER -
%0 Journal Article %A Alabau-Boussouira, Fatiha %A Prüss, Jan %A Zacher, Rico %T Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels %J Comptes Rendus. Mathématique %D 2009 %P 277-282 %V 347 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.01.005/ %R 10.1016/j.crma.2009.01.005 %G en %F CRMATH_2009__347_5-6_277_0
Alabau-Boussouira, Fatiha; Prüss, Jan; Zacher, Rico. Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels. Comptes Rendus. Mathématique, Tome 347 (2009) no. 5-6, pp. 277-282. doi : 10.1016/j.crma.2009.01.005. http://www.numdam.org/articles/10.1016/j.crma.2009.01.005/
[1] Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., Volume 328 (1999), pp. 1015-1020
[2] Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim., Volume 41 (2002), pp. 511-541
[3] Indirect internal damping of coupled systems, J. Evolution Equations, Volume 2 (2002), pp. 127-150
[4] Asymptotic stability of wave equations with memory and frictional boundary dampings, Appl. Math., Volume 35 (2008) no. 3, pp. 247-258
[5] Decay estimates for second order evolution equations with memory, J. Funct. Anal., Volume 254 (2008), pp. 1342-1372
[6] F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation
[7] An existence result for semilinear equations in viscoelasticity: the case of regular kernels (Fabrizio, M.; Lazzari, B.; Morro, A., eds.), Mathematical Models and Methods for Smart Materials, Series on Advances in Mathematics for Applied Sciences, vol. 62, World Scientific, 2002, pp. 343-354
[8] Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., Volume 42 (2003), pp. 1310-1324
[9] Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., Volume 37 (1970), pp. 297-308
[10] An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, Volume 7 (1970), pp. 554-569
[11] On the existence and the asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal., Volume 116 (1991), pp. 139-152
[12] Lyapunov functionals for reaction–diffusion equations with memory, Math. Methods Appl. Sci., Volume 28 (2005), pp. 1725-1735
[13] Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, 1994 (John Wiley and Sons, Ltd., Chichester)
[14] Contrôlabilité Exacte et Stabilisation de Systèmes Distribués I–II, Masson, Paris, 1988
[15] Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., Volume 52 (1994), pp. 628-648
[16] Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., Volume 59 (2001), pp. 557-578
[17] On wave equations with boundary dissipation of memory type, J. Integral Equations Appl., Volume 8 (1996), pp. 99-123
[18] Evolutionary Integral Equations and Applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993
[19] Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z., Volume 259 (2008), pp. 287-309
[20] R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint
[21] Maximal regularity of type for abstract parabolic Volterra equations, J. Evolution Equations, Volume 5 (2005), pp. 79-103
[22] R. Zacher, Quasilinear parabolic problems with nonlinear boundary conditions, Thesis, Martin-Luther-Universität Halle, 2003
Cité par Sources :
