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Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting
Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $
1. | Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China |
2. | School of Sciences, Beijing Forestry University, Beijing, 100083, China |
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.
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. |
[2] |
V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Academic Press, Inc., Boston, MA, 1993.
![]() |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Lagnese,
Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
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. |
[2] |
V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Academic Press, Inc., Boston, MA, 1993.
![]() |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Lagnese,
Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
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