Analysis of the energy decay of a viscoelasticity type equation

[1] S. Alinhac, P. Gérard, Opérateurs pseudo-differentiels et théoréme de Nash-Moser, Inter Éditions du CNRS, Meudon, France, 1991.10.1051/978-2-7598-0282-1Search in Google Scholar

[2] A. Atallah-Baraket, C. Fermanian Kammerer, High frequency analysis of solutions to the equation of viscoelasticity of Kelvin-Voight, J. Hyperbolic. Differ. Equ. 1 (2004), 789-812.10.1142/S0219891604000299Search in Google Scholar

[3] M.A. Ayadi, A. Bchatnia, M. Hamouda, S. Messaoudi, General decay in a Timoshenko-type system with thermoelasticity with second sound, Adv. Nonlinear Anal. 4 (2015), 263-284.10.1515/anona-2015-0038Search in Google Scholar

[4] D. Blanchard, O. Guib, Existence of a solution for a nonlinear system in thermo-viscoelasticity, Adv. Differential Equations 5 (2000), 1221-1252.10.57262/ade/1356651222Search in Google Scholar

[5] B. Bougherara, J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal. 4 (2015), 123-134.10.1515/anona-2015-0002Search in Google Scholar

[6] D. Brandon, I. Fonseca and P. Swart, Dynamics and oscillatory microstructure in a model of displacive phase transformations, Progress in partial differential equations: the Metz surveys, 3, 130144, Pitman Res. Notes Math. Ser., 314, Longman Sci. Tech., Harlow, 1994.Search in Google Scholar

[7] N. Burq, Mesures semi-classique et mesures de défaut, Sminaire N. Bourbaki, (1996-1997), exp. no 826, 167-195.Search in Google Scholar

[8] N. Burq, G. Lebeau, Mesures de défaut de compacité, application au systéme de Lamé, Ann. Scient. Ec. Norm. Sup. 34 (2001), 817-870.10.1016/S0012-9593(01)01078-3Search in Google Scholar

[9] E. Cordero, F. Nicola and L. Rodino, Microlocal analysis and applications, Advances in pseudo-differential operators, 117, Oper. Theory Adv. Appl., 155, Birkhuser, Basel, 2004.10.1007/978-3-0348-7840-1_1Search in Google Scholar

[10] C. Do, On the Dynamic Deformation of a Bar against an Obstacle. Variational Methods in the Mechanics of Solids, ed. S. Nemat Nasser, Northwestern University, USA ( Pergamon Press, 1978), 237-241.10.1016/B978-0-08-024728-1.50041-0Search in Google Scholar

[11] G. Francfort, Introduction to H-Measures and their applications, Progress in Nonlinear Differential Equations and Their Applications, Vol. 68, 85-110, Birkhuser Verlag Basel = Switzerland.10.1007/3-7643-7565-5_7Search in Google Scholar

[12] G. Francfort, P. Gérard, The wave equation on a thin domain : Energy Density and observability, Journal of Hyperbolic Differential Equation 1 (2004), 351-366.10.1142/S0219891604000159Search in Google Scholar

[13] G. Francfort, F. Murat, Oscillations and energy densities in the wave equation, Comm. Partial Differential Equations 17 (1992), 1785-1865.10.1080/03605309208820905Search in Google Scholar

[14] P. Gérard, Microlocal defect measures, Commun. Partial Differential Equations, 16 (1991), 1761-1794.10.1080/03605309108820822Open DOISearch in Google Scholar

[15] P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 133 (1996), 50-68.Search in Google Scholar

[16] M. Ghergu, V. R_adulescu, Nonlinear PDEs. Mathematical models in biology, chemistry and population genetics, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.10.1007/978-3-642-22664-9Search in Google Scholar

[17] M. S. Joshi, Introduction to Pseudo-Differential operators, arXiv: 9906155v1, (1999).Search in Google Scholar

[18] J. U. Kim, On the local regularity of solutions in linear viscoelasticity of several space dimensions, Trans. Amer . Math. Soc. 346 (1994), 359-398.10.1090/S0002-9947-1994-1270666-7Search in Google Scholar

[19] G. Lebeau, Equations des ondes amorties, Séminaire de l'école polytechnique, Expos XV (1994).Search in Google Scholar

[20] J. Le Rousseau, Analyse microlocale et semi classique. Applications au prolongement unique, et au contrôle des ondes et de la chaleur, unpublished manuscript.Search in Google Scholar

[21] A. Petrov, M. Schatzman, Viscolastodynamique monodimensionnelle avec conditions de Signorini, C.R. Acad. Sci. Paris Ser. I 334 (2002), 983-988.10.1016/S1631-073X(02)02399-3Search in Google Scholar

[22] M. Petrini, Behaviour of the energy density associated to a Kelvin-Voight model in viscoelasticity, Asymptot. Anal 34 (2003), 261-273.Search in Google Scholar

[23] V. Rădulescu, D. Repovš, Partial differential equations with variable exponents. Variational methods and qualitative analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.10.1201/b18601Search in Google Scholar

[24] L. Tartar, H-measures, a new approach for studying Homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc, Edinburgh Sect. A 115 (1990), 193-230.10.1017/S0308210500020606Search in Google Scholar

[25] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1934), 742-759.Search in Google Scholar