Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation

Emanuela R. S. Coelho; Valéria N. Domingos Cavalcanti; Vinicius A. Peralta

doi:10.3934/cpaa.2021055

Article Contents

2021, Volume 20, Issue 5: 1987-2020. Doi: 10.3934/cpaa.2021055

This issue Previous Article Spectral properties of ordinary differential operators admitting special decompositions Next Article A generalized complex Ginzburg-Landau equation: Global existence and stability results

Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation

Emanuela R. S. Coelho^1,,
Valéria N. Domingos Cavalcanti^2,3, , and
Vinicius A. Peralta^2,3,

1.
Department of Mathematics, State University of Paraíba, Campina Grande, PB 58429-500, Brazil
2.
Department of Mathematics, State University of Maringá, Maringá, PR 87020-900, Brazil
3.
Department of Mathematics, Federal University of Technology-Paraná, Cornélio Procópio, PR 86300-000, Brazil

^* Corresponding author
^* Corresponding author

Received: August 2020

Revised: February 2021

Early access: April 2021

Published: May 2021

Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant 304895/2003-2

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Abstract

We are concerned with the transmission problem of nonlinear viscoelastic waves in a heterogeneous medium, establishing the well-posedness of solutions and the exponential stability of the related energy functional. We introduce an auxiliary problem to prove the exponential stability and the proof combines an observability inequality and microlocal analysis tools.

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Mathematics Subject Classification: Primary: 35L05; Secondary: 35L53, 35B40.

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[1]	M. S. Alves, J. Muñoz Rivera, M. Sepúlveda and O. Vera Villagrán, The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (1992), 345-365. doi: 10.1137/130923233.
[2]	M. S. Alves, J. E. Muñoz Rivera, M. Sepúlveda, O. Vera Villagrán and M. Zegarra Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497. doi: 10.1002/mana.201200319.
[3]	J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.
[4]	L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wien. Ber., 70 (1874), 275-306.
[5]	L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wied. Ann., 5 (1878), 430-432.
[6]	N. Burq and P. Gérard, Contrôle Optimal des Équations Aux Dérivées Partielles, 2001. Available from: http://www.math.u-psud.fr/ burq/articles/coursX.pdf.
[7]	F. Cardoso and G. Vodev, Boundary stabilization of transmission problems, J. Math. Phys., 51(2010), 023512. doi: 10.1063/1.3277163.
[8]	M. Cavalcanti, L. Fatori and Ma To Fu, Attractors for wave equations with degenerate memory, J. Differ. Equ., 260 (2016), 56-83. doi: 10.1016/j.jde.2015.08.050.
[9]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva and A. Y. S. Franco, Exponential stability for the wave model with localized memory in a past history framework, J. Differ. Equ., 264 (2018), 6535-6584. doi: 10.1016/j.jde.2018.01.044.
[10]	M. M. Cavalcanti, E. R. S. Coelho and V. N. Domingos Cavalcanti, Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., 81 (2020), 621-650. doi: 10.1007/s00245-018-9514-9.
[11]	M. Conti, E. M. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216. doi: 10.1016/j.na.2013.08.015.
[12]	M. Conti, E. M. Marchini and V. Pata, Global attractors for nonlinear viscoelastic equations with memory, Commun. Pure Appl. Anal., 15 (2016), 1893-1913. doi: 10.3934/cpaa.2016021.
[13]	M. Conti, E. M. Marchini and V. Pata, Non classical diffusion with memory, Math. Meth. Appl. Sci., 38 (2015), 948-958. doi: 10.1002/mma.3120.
[14]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.
[15]	V. Danese, P. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904. doi: 10.3934/dcds.2015.35.2881.
[16]	T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41. doi: 10.1016/j.anihpc.2006.07.005.
[17]	M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3.
[18]	H. D. Fernandéz Sare and J. E. Muñoz Rivera, Analyticity of transmission problem to thermoelastic plates, Quart. Appl. Math., 69 (2011), 1-13. doi: 10.1090/S0033-569X-2010-01187-6.
[19]	L. Gagnon, Sufficient Conditions for the Controllability of Wave Equations with a Transmission Condition at the Interface, preprint, arXiv: 1711.00448.
[20]	P. Gérard, Microlocal defect measures, Commun. Partial Differ. Equ., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.
[21]	C. Giorgi, J. E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99. doi: 10.1006/jmaa.2001.7437.
[22]	M. Grasselli and V. Pata, Uniform attractors of non autonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (Eds. A. Lorenzi, B. Ruf), Birkhauser Verlag Basel/Switzerland, 50 (2002), 155–178. doi: 10.1007/978-3-0348-8221-7_9.
[23]	A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485. doi: 10.1016/j.nonrwa.2011.08.004.
[24]	Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ., 257 (2014), 3778-3812. doi: 10.1016/j.jde.2014.07.009.
[25]	M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467.
[26]	J. E. Lagnese, Boundary controllability in problems of transmission for a class of Second order hyperbolic systems, ESAIM: Control Optim. Calc. Var., 2 (1997), 343-357. doi: 10.1051/cocv:1997112.
[27]	J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Coll. RMA, vol.8, Masson, Paris, 1988.
[28]	W. Liu, Stabilization and controllability for the transmission wave equation, IEEE Tran. Auto. Control, 46 (2001), 1900-1907. doi: 10.1109/9.975473.
[29]	K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, ZAMP, 53 (2002), 265-280. doi: 10.1007/s00033-002-8155-6.
[30]	W. Liu and G. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 57 (1998), 305-327. doi: 10.1017/S0004972700031683.
[31]	S. Nicaise, Boundary exact controllability of interface problems with singularities I: addition of the coefficients of singularities, SIAM J. Control Optim., 34 (1996), 1512-1532. doi: 10.1137/S0363012995282103.
[32]	S. Nicaise, Boundary exact controllability of interface problems with singularities II: addition of internal controls, SIAM J. Control Optim., 35 (1997), 585-603. doi: 10.1137/S0363012995292032.
[33]	T. Özsari, V. K. Kalantarov and I. Lasiecka, Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differ. Equ., 251 (2011), 1841-1863. doi: 10.1016/j.jde.2011.04.003.
[34]	V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333–360.,
[35]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[36]	J. E. Muñoz Rivera and M. G. Naso, About asymptotic behavior for a transmission problem in hyperbolic thermoelasticity, Acta Appl. Math., 99 (2007), 1-27. doi: 10.1007/s10440-007-9152-8.
[37]	J. E. Muñoz Rivera and H. P. Oquendo, The transmission problem of viscoelastic waves, Acta Appl. Math., 62 (2000), 1-21. doi: 10.1023/A:1006449032100.
[38]	J. Simon, Compact Sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[39]	V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356. doi: 10.1007/BF02418820.
[40]	V. Volterra, LeÇons Sur Les Fonctions De Lignes, Gauthier-Villars, Paris, 1913.