American Institute of Mathematical Sciences

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May  2021, 20(5): 1987-2020. doi: 10.3934/cpaa.2021055

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

Received  August 2020 Revised  February 2021 Published  May 2021 Early access  April 2021

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

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.

Keywords: Nonlinear wave equation, transmission problem, viscoelastic effect, viscoelastic wave equation, exponential stability.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L53, 35B40.
Citation: Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1987-2020. doi: 10.3934/cpaa.2021055
References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wien. Ber., 70 (1874), 275-306.   Google Scholar

[5]

L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wied. Ann., 5 (1878), 430-432.   Google Scholar

[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. Google Scholar

[7]

F. Cardoso and G. Vodev, Boundary stabilization of transmission problems, J. Math. Phys., 51(2010), 023512. doi: 10.1063/1.3277163.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Gagnon, Sufficient Conditions for the Controllability of Wave Equations with a Transmission Condition at the Interface, preprint, arXiv: 1711.00448. Google Scholar

[20]

P. Gérard, Microlocal defect measures, Commun. Partial Differ. Equ., 16 (1991), 1761-1794.  doi: 10.1080/03605309108820822.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Coll. RMA, vol.8, Masson, Paris, 1988. Google Scholar

[28]

W. Liu, Stabilization and controllability for the transmission wave equation, IEEE Tran. Auto. Control, 46 (2001), 1900-1907.  doi: 10.1109/9.975473.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333–360., Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.  doi: 10.1007/BF02418820.  Google Scholar

[40]

V. Volterra, LeÇons Sur Les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wien. Ber., 70 (1874), 275-306.   Google Scholar

[5]

L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Wied. Ann., 5 (1878), 430-432.   Google Scholar

[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. Google Scholar

[7]

F. Cardoso and G. Vodev, Boundary stabilization of transmission problems, J. Math. Phys., 51(2010), 023512. doi: 10.1063/1.3277163.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Gagnon, Sufficient Conditions for the Controllability of Wave Equations with a Transmission Condition at the Interface, preprint, arXiv: 1711.00448. Google Scholar

[20]

P. Gérard, Microlocal defect measures, Commun. Partial Differ. Equ., 16 (1991), 1761-1794.  doi: 10.1080/03605309108820822.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Coll. RMA, vol.8, Masson, Paris, 1988. Google Scholar

[28]

W. Liu, Stabilization and controllability for the transmission wave equation, IEEE Tran. Auto. Control, 46 (2001), 1900-1907.  doi: 10.1109/9.975473.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333–360., Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.  doi: 10.1007/BF02418820.  Google Scholar

[40]

V. Volterra, LeÇons Sur Les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. Google Scholar

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