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June  2022, 15(6): 1339-1354. doi: 10.3934/dcdss.2022090

The lack of exponential stability for a weakly coupled wave equations through a variable density term

Université de Monastir, Faculté des Sciences de Monastir, Analyse et Contrôle des EDP, UR13ES64, Monastir, Tunisie

Received  January 2022 Revised  March 2022 Published  June 2022 Early access  April 2022

In this paper, we consider a system of two wave equations coupled through zero order terms. One of these equations has an internal damping, and the other has a boundary damping. We investigate stability properties of the system according to the variable strings densities. Indeed, our main result is to show that the corresponding model is not exponentially stable using a spectral theory which forms the center of this work. Otherwise, we establish a polynomial energy decay rate of type $ \frac{1}{\sqrt{t}}. $

Citation: Monia Bel Hadj Salah. The lack of exponential stability for a weakly coupled wave equations through a variable density term. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1339-1354. doi: 10.3934/dcdss.2022090
References:
[1]

F. AbdallahS. NicaiseJ. Valein and A. Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications, ESAIM Control Optim. Calc. Var., 19 (2013), 844-887.  doi: 10.1051/cocv/2012036.

[2]

M. Afilal and F. Ammar-Khodja, Stability of coupled second order equations, Comput. Appl. Math., 19 (2000), 91–107,126.

[3]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris SØ©r. I Math., 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.

[4]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.

[5]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541.  doi: 10.1137/S0363012901385368.

[6]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582.  doi: 10.1051/cocv/2011106.

[7]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptot. Anal., 28 (2001), 215-240. 

[8]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[9]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[10]

K. AmmariD. Mercier and V. Régnier, Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications, J. Differential Equations, 259 (2015), 6923-6959.  doi: 10.1016/j.jde.2015.08.017.

[11]

K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, 2124, Springer, Cham, 2015. doi: 10.1007/978-3-319-10900-8.

[12]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, doi: 10.1051/cocv:2001114.

[13]

K. AmmariM. Tucsnak and A. Henrot, Optimal location of the actuator for the pointwise stabilization of a string, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386.  doi: 10.1016/S0764-4442(00)00113-0.

[14]

F. Ammar-Khodja and A. Bader, Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851.  doi: 10.1137/S0363012900366613.

[15]

F. Ammar-Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamical controllers, Dynam. Contin. Discrete Impuls. Systems, 7 (2000), 207-222. 

[16]

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.

[17]

M. Bassam, D. Mercier, S. Nicaise and A. Wehbe, Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane, ZAMM Z. Angew. Math. Mech., 96 (2016), 916-938. doi: 10.1002/zamm.201500172.

[18]

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.

[19]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[20]

M. Bel Hadj Salah, Stabilization of weakly coupled wave equations through a density term, Eur. J. Control, 58 (2021), 315-326.  doi: 10.1016/j.ejcon.2020.07.010.

[21]

A. Benaddi, Optimal energy decay rate of coupled wave equations, Port. Math. (N.S.), 61 (2004), 81-96. 

[22]

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.

[23]

S. Chai, Uniform decay rate for the transmission wave equations with variable coefficients, J. Syst. Sci. Complex., 24 (2011), 253-260.  doi: 10.1007/s11424-011-8009-4.

[24]

S. Chai and K. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A., 26 (2005), 605-612. 

[25]

G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Volume I: Analysis, Estimation, Attenuation, and Design. CRC Press, Inc. Boca Raton, FL, 1993.

[26]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.

[27]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212. 

[28]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.  doi: 10.1215/S0012-7094-97-08614-2.

[29]

W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17-34.  doi: 10.1137/S0036139996314106.

[30]

W. Liu and G. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.

[31]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881.  doi: 10.1016/j.jmaa.2007.02.021.

[32]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, 398 Research Notes in Mathematics, Champman-Hall/CRC, 1999.

[33]

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.

[34]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319.

[35]

Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022.

[36]

A. S. Marcus, Introduction to the spectral theory of polynomial pencils, Trans. Math. Monogr., 71, Providence, RI, 1988. doi: 10.1090/mmono/071.

[37]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM: COCV., 16 (2010), 420-456.  doi: 10.1051/cocv/2009007.

[38]

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

[39]

J. Prüss, On the spectrum of C-0 semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[40]

B. Rao, On the sensitivity of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations, Z. Angew. Math. Phys., 70 (2019), Paper No. 75. doi: 10.1007/s00033-019-1110-6.

[41]

B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent, J. Evol. Equ., 5 (2005), 137-152.  doi: 10.1007/s00028-005-0171-5.

[42]

J. RauchX. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006.

[43]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

show all references

References:
[1]

F. AbdallahS. NicaiseJ. Valein and A. Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications, ESAIM Control Optim. Calc. Var., 19 (2013), 844-887.  doi: 10.1051/cocv/2012036.

[2]

M. Afilal and F. Ammar-Khodja, Stability of coupled second order equations, Comput. Appl. Math., 19 (2000), 91–107,126.

[3]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris SØ©r. I Math., 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.

[4]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.

[5]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541.  doi: 10.1137/S0363012901385368.

[6]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582.  doi: 10.1051/cocv/2011106.

[7]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptot. Anal., 28 (2001), 215-240. 

[8]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[9]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[10]

K. AmmariD. Mercier and V. Régnier, Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications, J. Differential Equations, 259 (2015), 6923-6959.  doi: 10.1016/j.jde.2015.08.017.

[11]

K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, 2124, Springer, Cham, 2015. doi: 10.1007/978-3-319-10900-8.

[12]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, doi: 10.1051/cocv:2001114.

[13]

K. AmmariM. Tucsnak and A. Henrot, Optimal location of the actuator for the pointwise stabilization of a string, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386.  doi: 10.1016/S0764-4442(00)00113-0.

[14]

F. Ammar-Khodja and A. Bader, Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851.  doi: 10.1137/S0363012900366613.

[15]

F. Ammar-Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamical controllers, Dynam. Contin. Discrete Impuls. Systems, 7 (2000), 207-222. 

[16]

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.

[17]

M. Bassam, D. Mercier, S. Nicaise and A. Wehbe, Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane, ZAMM Z. Angew. Math. Mech., 96 (2016), 916-938. doi: 10.1002/zamm.201500172.

[18]

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.

[19]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[20]

M. Bel Hadj Salah, Stabilization of weakly coupled wave equations through a density term, Eur. J. Control, 58 (2021), 315-326.  doi: 10.1016/j.ejcon.2020.07.010.

[21]

A. Benaddi, Optimal energy decay rate of coupled wave equations, Port. Math. (N.S.), 61 (2004), 81-96. 

[22]

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.

[23]

S. Chai, Uniform decay rate for the transmission wave equations with variable coefficients, J. Syst. Sci. Complex., 24 (2011), 253-260.  doi: 10.1007/s11424-011-8009-4.

[24]

S. Chai and K. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A., 26 (2005), 605-612. 

[25]

G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Volume I: Analysis, Estimation, Attenuation, and Design. CRC Press, Inc. Boca Raton, FL, 1993.

[26]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.

[27]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212. 

[28]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.  doi: 10.1215/S0012-7094-97-08614-2.

[29]

W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17-34.  doi: 10.1137/S0036139996314106.

[30]

W. Liu and G. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.

[31]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881.  doi: 10.1016/j.jmaa.2007.02.021.

[32]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, 398 Research Notes in Mathematics, Champman-Hall/CRC, 1999.

[33]

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.

[34]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319.

[35]

Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022.

[36]

A. S. Marcus, Introduction to the spectral theory of polynomial pencils, Trans. Math. Monogr., 71, Providence, RI, 1988. doi: 10.1090/mmono/071.

[37]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM: COCV., 16 (2010), 420-456.  doi: 10.1051/cocv/2009007.

[38]

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

[39]

J. Prüss, On the spectrum of C-0 semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[40]

B. Rao, On the sensitivity of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations, Z. Angew. Math. Phys., 70 (2019), Paper No. 75. doi: 10.1007/s00033-019-1110-6.

[41]

B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent, J. Evol. Equ., 5 (2005), 137-152.  doi: 10.1007/s00028-005-0171-5.

[42]

J. RauchX. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006.

[43]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

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