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June  2021, 10(2): 321-331. doi: 10.3934/eect.2020068

Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $

Bei Gong 1, , Zhen-Hu Ning 1, and  Fengyan Yang 2,,

1. 

Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China
2. 

School of Sciences, Beijing Forestry University, Beijing, 100083, China

* Corresponding author: Fengyan Yang

Received  December 2019 Revised  February 2020 Published  June 2020

In this article, we consider the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n(n\ge 3) $. By virtue of the Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments, we obtain some stability result of the transmission wave/plate system under suitable geometric conditions.

Keywords: stabilization, wave/plate equation, variable coefficients, Riemannian geometry.
Mathematics Subject Classification: Primary: 93C20; Secondary: 93D20.
Citation: Bei Gong, Zhen-Hu Ning, Fengyan Yang. Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $. Evolution Equations & Control Theory, 2021, 10 (2) : 321-331. doi: 10.3934/eect.2020068
References:
[1]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.  Google Scholar

[2] V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Academic Press, Inc., Boston, MA, 1993.   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.   Google Scholar

[4]

C. A. Bortot, M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Piccione, Exponential asymptotic stability for the Klein-Gordon equation on non-compact Riemannian manifolds, Appl. Math. Optim., 78 (2018), 219-265.  doi: 10.1007/s00245-017-9405-5.  Google Scholar

[5]

N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains, Commun. Contemp. Math., 18 (2016), 1650012.  doi: 10.1142/S0219199716500127.  Google Scholar

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J. M. Bouclet and J. Royer, Local energy decay for the damped wave equation, J. Funct. Anal., 266 (2014), 4538-4615.  doi: 10.1016/j.jfa.2014.01.028.  Google Scholar

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V. Barbu, I. Lasiecka and A. M. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana Univ. Math. J., 56 (2007), 995-1021.  doi: 10.1512/iumj.2007.56.2990.  Google Scholar

[8]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. École Norm. Sup., 36 (2003), 525-551.  doi: 10.1016/S0012-9593(03)00021-1.  Google Scholar

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S. J. Feng and D. X. Feng, Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin. (Engl. Ser.), 20 (2004), 1057-1072.  doi: 10.1007/s10114-004-0394-3.  Google Scholar

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B. Gong, F. Y. Yang and X. Zhao, Stabilization of the transmission wave/plate equation with variable coefficients, J. Math. Anal. Appl., 455 (2017), 947-962.  doi: 10.1016/j.jmaa.2017.06.014.  Google Scholar

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A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[12]

I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.  doi: 10.1080/03605309908821495.  Google Scholar

[13]

J. Lagnese, Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85.   Google Scholar

[14]

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

[15]

Y. X. Liu and P. F. Yao, Energy decay rate of the wave equations on Riemannian manifolds with critical potential, Appl. Math. Optim., 78 (2018), 61-101.  doi: 10.1007/s00245-017-9399-z.  Google Scholar

[16]

C. Morawetz, Time decay for nonlinear Klein-Gordon equations, Proc. Roy. Soc. London Ser. A, 306 (1968), 503-518.  doi: 10.1098/rspa.1968.0151.  Google Scholar

[17]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.  Google Scholar

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M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417.  doi: 10.1007/BF01444231.  Google Scholar

[19]

M. Nakao, Energy decay for the linear and semilinear wave equation in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797.  doi: 10.1007/s002090100275.  Google Scholar

[20]

M. Nakao, Existence of global solutions for the Kirchhoff-type quasilinear wave equation in exterior domains with a half-linear dissipation, Kyushu J. Math., 58 (2004), 373-391.  doi: 10.2206/kyushujm.58.373.  Google Scholar

[21]

Z. H. Ning, F. Y. Yang and X. P. Zhao, Escape metrics and its applications, preprint, 2018, arXiv: 1811.12668. Google Scholar

[22]

M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc., 354 (2002), 3621-3637.  doi: 10.1090/S0002-9947-02-03034-9.  Google Scholar

[23]

M. Slemrod, Weak asymptotic decay via a related invariance principle for a wave equation with nonlinear, nonmonotone damping, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 87-97.  doi: 10.1017/S0308210500023970.  Google Scholar

[24]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal., 41 (2000), 891-905.  doi: 10.1016/S0362-546X(98)00317-4.  Google Scholar

[25]

G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in ${\mathbb{R}}^n$, Indiana Univ. Math. J., 56 (2007), 389-416.  doi: 10.1512/iumj.2007.56.2963.  Google Scholar

[26]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[27]

L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations, 145 (1998), 502-524.  doi: 10.1006/jdeq.1998.3416.  Google Scholar

[28]

P. F. Yao, Energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 (2010), 59-70.  doi: 10.1007/s11401-008-0421-2.  Google Scholar

[29]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Contemporary Mathematics, Vol. 268, Amer. Math. Soc., Providence, RI, (2000), 383–406. doi: 10.1090/conm/268/04320.  Google Scholar

[30] P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics. A Differential Geometric Approach, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b11042.  Google Scholar
[31]

P. F. Yao, Y. X. Liu and J. Li, Decay rates of the hyperbolic equation in an exterior domain with half-linear and nonlinear boundary dissipations, J. Syst. Sci. Complex, 29 (2016), 657-680.  doi: 10.1007/s11424-015-4233-7.  Google Scholar

[32]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.  Google Scholar

show all references

References:
[1]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.  Google Scholar

[2] V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Academic Press, Inc., Boston, MA, 1993.   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.   Google Scholar

[4]

C. A. Bortot, M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Piccione, Exponential asymptotic stability for the Klein-Gordon equation on non-compact Riemannian manifolds, Appl. Math. Optim., 78 (2018), 219-265.  doi: 10.1007/s00245-017-9405-5.  Google Scholar

[5]

N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains, Commun. Contemp. Math., 18 (2016), 1650012.  doi: 10.1142/S0219199716500127.  Google Scholar

[6]

J. M. Bouclet and J. Royer, Local energy decay for the damped wave equation, J. Funct. Anal., 266 (2014), 4538-4615.  doi: 10.1016/j.jfa.2014.01.028.  Google Scholar

[7]

V. Barbu, I. Lasiecka and A. M. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana Univ. Math. J., 56 (2007), 995-1021.  doi: 10.1512/iumj.2007.56.2990.  Google Scholar

[8]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. École Norm. Sup., 36 (2003), 525-551.  doi: 10.1016/S0012-9593(03)00021-1.  Google Scholar

[9]

S. J. Feng and D. X. Feng, Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin. (Engl. Ser.), 20 (2004), 1057-1072.  doi: 10.1007/s10114-004-0394-3.  Google Scholar

[10]

B. Gong, F. Y. Yang and X. Zhao, Stabilization of the transmission wave/plate equation with variable coefficients, J. Math. Anal. Appl., 455 (2017), 947-962.  doi: 10.1016/j.jmaa.2017.06.014.  Google Scholar

[11]

A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[12]

I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.  doi: 10.1080/03605309908821495.  Google Scholar

[13]

J. Lagnese, Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85.   Google Scholar

[14]

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

[15]

Y. X. Liu and P. F. Yao, Energy decay rate of the wave equations on Riemannian manifolds with critical potential, Appl. Math. Optim., 78 (2018), 61-101.  doi: 10.1007/s00245-017-9399-z.  Google Scholar

[16]

C. Morawetz, Time decay for nonlinear Klein-Gordon equations, Proc. Roy. Soc. London Ser. A, 306 (1968), 503-518.  doi: 10.1098/rspa.1968.0151.  Google Scholar

[17]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.  Google Scholar

[18]

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417.  doi: 10.1007/BF01444231.  Google Scholar

[19]

M. Nakao, Energy decay for the linear and semilinear wave equation in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797.  doi: 10.1007/s002090100275.  Google Scholar

[20]

M. Nakao, Existence of global solutions for the Kirchhoff-type quasilinear wave equation in exterior domains with a half-linear dissipation, Kyushu J. Math., 58 (2004), 373-391.  doi: 10.2206/kyushujm.58.373.  Google Scholar

[21]

Z. H. Ning, F. Y. Yang and X. P. Zhao, Escape metrics and its applications, preprint, 2018, arXiv: 1811.12668. Google Scholar

[22]

M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc., 354 (2002), 3621-3637.  doi: 10.1090/S0002-9947-02-03034-9.  Google Scholar

[23]

M. Slemrod, Weak asymptotic decay via a related invariance principle for a wave equation with nonlinear, nonmonotone damping, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 87-97.  doi: 10.1017/S0308210500023970.  Google Scholar

[24]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal., 41 (2000), 891-905.  doi: 10.1016/S0362-546X(98)00317-4.  Google Scholar

[25]

G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in ${\mathbb{R}}^n$, Indiana Univ. Math. J., 56 (2007), 389-416.  doi: 10.1512/iumj.2007.56.2963.  Google Scholar

[26]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[27]

L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations, 145 (1998), 502-524.  doi: 10.1006/jdeq.1998.3416.  Google Scholar

[28]

P. F. Yao, Energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 (2010), 59-70.  doi: 10.1007/s11401-008-0421-2.  Google Scholar

[29]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Contemporary Mathematics, Vol. 268, Amer. Math. Soc., Providence, RI, (2000), 383–406. doi: 10.1090/conm/268/04320.  Google Scholar

[30] P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics. A Differential Geometric Approach, CRC Press, Boca Raton, FL, 2011.  doi: 10.1201/b11042.  Google Scholar
[31]

P. F. Yao, Y. X. Liu and J. Li, Decay rates of the hyperbolic equation in an exterior domain with half-linear and nonlinear boundary dissipations, J. Syst. Sci. Complex, 29 (2016), 657-680.  doi: 10.1007/s11424-015-4233-7.  Google Scholar

[32]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.  Google Scholar

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