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December  2021, 11(4): 885-904. doi: 10.3934/mcrf.2020050

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Ali Wehbe 2,, , Rayan Nasser 1,2, and  Nahla Noun 1,2,

1. 

Université de Bretagne Occidentale, LMBA, Brest, France, Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon
2. 

Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon

* Corresponding author: Ali Wehbe

Received  February 2020 Revised  September 2020 Published  December 2021 Early access  December 2020

Fund Project: The first author is supported by the CNRS and the LAMA laboratory of Mathematics of the Université Savoie Mont Blanc

We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type $ t^{-1} $ for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type $ t^{-2/5} $ if the strip is localized near the boundary. Else, it's of type $ t^{-1/3} $. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [31], [28], [19] and include in particular an example where the damping region is localized faraway from the boundary. Note that part of the results of this paper was announced in [22].

Keywords: Wave equation, transmission problem, local Kelvin-Voigt damping, geometric conditions, stabilization.
Mathematics Subject Classification: Primary: 35B35, 93B52; Secondary: 93C20.
Citation: Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2021, 11 (4) : 885-904. doi: 10.3934/mcrf.2020050
References:
[1]

K. Ammari, F. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal., 236 (2020), 577-601.  doi: 10.1007/s00205-019-01476-4.  Google Scholar

[2]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[3]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[6]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[7]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.  doi: 10.3233/ASY-1997-14203.  Google Scholar

[8]

M. Cavalcanti, V. D. Cavalcanti and L. Tebou, Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms, Electron. J. Differential Equations, (2017), 18pp.  Google Scholar

[9]

S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.  Google Scholar

[10]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces., Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[11]

F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.  Google Scholar

[12]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[13]

G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 73–109.  Google Scholar

[14]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués: Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988.  Google Scholar

[15]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.  Google Scholar

[16]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.  Google Scholar

[17]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.  Google Scholar

[18]

K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.  Google Scholar

[19]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.  doi: 10.1007/s00033-005-0029-2.  Google Scholar

[20]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[21]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.  Google Scholar

[22]

R. Nasser, N. Noun and A. Wehbe, Stabilization of the wave equations with localized Kelvin-Voigt type damping under optimal geometric conditions, C. R. Math. Acad. Sci. Paris, 357 (2019), 272-277.  doi: 10.1016/j.crma.2019.01.005.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.  Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[25]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[26]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[27]

R. Stahn, Optimal decay rate for the wave equation on a square with constant damping on a strip, Z. Angew. Math. Phys., 68 (2017), 10pp. doi: 10.1007/s00033-017-0781-0.  Google Scholar

[28]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.  doi: 10.1016/j.crma.2012.06.005.  Google Scholar

[29]

Q. Zhang, Exponential stability of an elastic string with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 61 (2010), 1009-1015.  doi: 10.1007/s00033-010-0064-5.  Google Scholar

[30]

Q. Zhang, On the lack of exponential stability for an elastic-viscoelastic waves interaction system, Nonlinear Anal. Real World Appl., 37 (2017), 387-411.  doi: 10.1016/j.nonrwa.2017.02.019.  Google Scholar

[31]

Q. Zhang, Polynomial decay of an elastic/viscoelastic waves interaction system, Z. Angew. Math. Phys., 69 (2018), 10pp. doi: 10.1007/s00033-018-0981-2.  Google Scholar

show all references

References:
[1]

K. Ammari, F. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal., 236 (2020), 577-601.  doi: 10.1007/s00205-019-01476-4.  Google Scholar

[2]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[3]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[6]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[7]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.  doi: 10.3233/ASY-1997-14203.  Google Scholar

[8]

M. Cavalcanti, V. D. Cavalcanti and L. Tebou, Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms, Electron. J. Differential Equations, (2017), 18pp.  Google Scholar

[9]

S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.  Google Scholar

[10]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces., Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[11]

F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.  Google Scholar

[12]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[13]

G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 73–109.  Google Scholar

[14]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués: Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988.  Google Scholar

[15]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.  Google Scholar

[16]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.  Google Scholar

[17]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.  Google Scholar

[18]

K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.  Google Scholar

[19]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.  doi: 10.1007/s00033-005-0029-2.  Google Scholar

[20]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[21]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.  Google Scholar

[22]

R. Nasser, N. Noun and A. Wehbe, Stabilization of the wave equations with localized Kelvin-Voigt type damping under optimal geometric conditions, C. R. Math. Acad. Sci. Paris, 357 (2019), 272-277.  doi: 10.1016/j.crma.2019.01.005.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.  Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[25]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[26]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[27]

R. Stahn, Optimal decay rate for the wave equation on a square with constant damping on a strip, Z. Angew. Math. Phys., 68 (2017), 10pp. doi: 10.1007/s00033-017-0781-0.  Google Scholar

[28]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.  doi: 10.1016/j.crma.2012.06.005.  Google Scholar

[29]

Q. Zhang, Exponential stability of an elastic string with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 61 (2010), 1009-1015.  doi: 10.1007/s00033-010-0064-5.  Google Scholar

[30]

Q. Zhang, On the lack of exponential stability for an elastic-viscoelastic waves interaction system, Nonlinear Anal. Real World Appl., 37 (2017), 387-411.  doi: 10.1016/j.nonrwa.2017.02.019.  Google Scholar

[31]

Q. Zhang, Polynomial decay of an elastic/viscoelastic waves interaction system, Z. Angew. Math. Phys., 69 (2018), 10pp. doi: 10.1007/s00033-018-0981-2.  Google Scholar

Figure 1.  Elastic-viscoelastic waves interaction models satisfying the assumption (A1)
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Figure 2.  A model satisfying assumption (A2)
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Figure 3.  A model satisfying both (A1) and (A2)
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Figure 4.  A square model with local viscoelastic strip not satisfying any geometry cited before
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Figure 5.  A model satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $
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Figure 6.  A model not satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $
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