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On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation
1. | Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
2. | School of Mathematical Sciences University of Chinese Academy of Sciences, Beijing 100049, China |
Long time behavior of a semilinear wave equation with variable coefficients with nonlinear boundary dissipation is considered. It is shown that the existence of global and compact attractors depends on the curvature properties of a Riemannian metric given by the variable coefficients.
References:
[1] |
I. Chueshov,
Strong solutions and attractors for von Karman equations, Math USSR Sbornik, 69 (1991), 25-36.
doi: 10.1070/SM1991v069n01ABEH001230. |
[2] |
I. Chueshov, M. Eller and I. Lasiecka,
On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. PDE., 27 (2002), 1901-1951.
doi: 10.1081/PDE-120016132. |
[3] |
I. Chueshov and I. Lasiecka,
Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Differ. Equ., 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[4] |
I. Chueshov, M. Eller and I. Lasiecka,
Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Commun. Partial Differ. Equ., 29 (2005), 1847-1876.
doi: 10.1081/PDE-200040203. |
[5] |
I. Chueshov and I. Lasiecka,
On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Differ. Equ., 36 (2010), 67-99.
doi: 10.1080/03605302.2010.484472. |
[6] |
I. Chueshov, Igor and I. Lasiecka,
Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differ. Equ., 198 (2004), 196-231.
doi: 10.1016/j.jde.2003.08.008. |
[7] |
I. Chueshov and I. Lasiecka,
Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.
doi: 10.1007/s00245-007-9031-8. |
[8] |
E. Fereisel,
Global attractors for semilinear damped wave equations with supercritical exponent, J. Differ. Equ., 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[9] |
B. Francesca and D. Toundykov,
Finite dimensional attractor for a composite system of wave/plate equations with localised damping, Nonlinearity, 23 (2010), 2271-2306.
doi: 10.1088/0951-7715/23/9/011. |
[10] |
J. Ghidaglia and R. Temam,
Regularity of the solutions of second order evolution equations and their attractors, Annali Della Scuola Normale Superiore di Pisa, 14 (1987), 485-511.
|
[11] |
J. Ghidaglia and R. Temam,
Attractors of damped nonlinear hyperbolic equations, J. Math. Pure Appl., 66 (1987), 273-319.
|
[12] |
J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988.
doi: 10.1090/surv/025. |
[13] |
A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, In MathematicalReports, Harwood Gordon Breach, NewYork, 1987. |
[14] |
I. Lasiecka,
Finite-Dimensionality of Attractors Associated with von Karman Plate Equations and Boundary Damping, J. Differ. Equ., 117 (1995), 357-389.
doi: 10.1006/jdeq.1995.1057. |
[15] |
I. Lasiecka, Local and global compact attractors arising in nonlinear elasticity, the case of noncompact nonlinearity and nonlinear dissipation, J. Math. Anal. Appl., 196 (1995), 332–C360.
doi: 10.1006/jmaa.1995.1413. |
[16] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of a semilinear wave equation with nonlinear boundary dissipation, Differ. Integral Equ., 6 (1993), 507-533.
|
[17] |
I. Lasiecka and R. Triggiani,
Uniform stabilization of the wave equation with dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.
doi: 10.1007/BF01182480. |
[18] |
I. Lasiecka, I. Chueshov and F. Bucci,
Global attractor for a composite system of nonlinear wave and plate equations, Commun. Pure Appl. Anal., 6 (2012), 113-140.
doi: 10.3934/cpaa.2007.6.113. |
[19] |
I. Lasiecka and I. Chueshov,
Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2012), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[20] |
R. Showalter, Monotone operators in banach spaces and nonlinear partial differential equations, AMS, Providence, 1997.
doi: 10.1090/surv/049. |
[21] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata Serie, 148 (1987), 65-96.
doi: 10.1007/BF01762360. |
[22] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[23] |
R. Triggian and P. F. Yao,
Carleman estimates with no lower-order terms for general riemann wave equations. global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375.
doi: 10.1007/s00245-002-0751-5. |
[24] |
P. F. Yao,
On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.
doi: 10.1137/S0363012997331482. |
[25] |
P. F. Yao,
Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61 (2010), 191-233.
doi: 10.1007/s00245-009-9088-7. |
[26] |
P. F. Yao,
Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differ. Equ., 241 (2007), 62-93.
doi: 10.1016/j.jde.2007.06.014. |
[27] |
P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics. A differential geometric approach. Chapman Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.
doi: 10.1201/b11042.![]() ![]() ![]() |
[28] |
Z. F. Zhang and P. F. Yao,
Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks, SIAM J. Control Optim., 47 (2008), 2044-2077.
doi: 10.1137/070679454. |
show all references
References:
[1] |
I. Chueshov,
Strong solutions and attractors for von Karman equations, Math USSR Sbornik, 69 (1991), 25-36.
doi: 10.1070/SM1991v069n01ABEH001230. |
[2] |
I. Chueshov, M. Eller and I. Lasiecka,
On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. PDE., 27 (2002), 1901-1951.
doi: 10.1081/PDE-120016132. |
[3] |
I. Chueshov and I. Lasiecka,
Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Differ. Equ., 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[4] |
I. Chueshov, M. Eller and I. Lasiecka,
Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Commun. Partial Differ. Equ., 29 (2005), 1847-1876.
doi: 10.1081/PDE-200040203. |
[5] |
I. Chueshov and I. Lasiecka,
On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Differ. Equ., 36 (2010), 67-99.
doi: 10.1080/03605302.2010.484472. |
[6] |
I. Chueshov, Igor and I. Lasiecka,
Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differ. Equ., 198 (2004), 196-231.
doi: 10.1016/j.jde.2003.08.008. |
[7] |
I. Chueshov and I. Lasiecka,
Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.
doi: 10.1007/s00245-007-9031-8. |
[8] |
E. Fereisel,
Global attractors for semilinear damped wave equations with supercritical exponent, J. Differ. Equ., 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[9] |
B. Francesca and D. Toundykov,
Finite dimensional attractor for a composite system of wave/plate equations with localised damping, Nonlinearity, 23 (2010), 2271-2306.
doi: 10.1088/0951-7715/23/9/011. |
[10] |
J. Ghidaglia and R. Temam,
Regularity of the solutions of second order evolution equations and their attractors, Annali Della Scuola Normale Superiore di Pisa, 14 (1987), 485-511.
|
[11] |
J. Ghidaglia and R. Temam,
Attractors of damped nonlinear hyperbolic equations, J. Math. Pure Appl., 66 (1987), 273-319.
|
[12] |
J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988.
doi: 10.1090/surv/025. |
[13] |
A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, In MathematicalReports, Harwood Gordon Breach, NewYork, 1987. |
[14] |
I. Lasiecka,
Finite-Dimensionality of Attractors Associated with von Karman Plate Equations and Boundary Damping, J. Differ. Equ., 117 (1995), 357-389.
doi: 10.1006/jdeq.1995.1057. |
[15] |
I. Lasiecka, Local and global compact attractors arising in nonlinear elasticity, the case of noncompact nonlinearity and nonlinear dissipation, J. Math. Anal. Appl., 196 (1995), 332–C360.
doi: 10.1006/jmaa.1995.1413. |
[16] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of a semilinear wave equation with nonlinear boundary dissipation, Differ. Integral Equ., 6 (1993), 507-533.
|
[17] |
I. Lasiecka and R. Triggiani,
Uniform stabilization of the wave equation with dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.
doi: 10.1007/BF01182480. |
[18] |
I. Lasiecka, I. Chueshov and F. Bucci,
Global attractor for a composite system of nonlinear wave and plate equations, Commun. Pure Appl. Anal., 6 (2012), 113-140.
doi: 10.3934/cpaa.2007.6.113. |
[19] |
I. Lasiecka and I. Chueshov,
Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2012), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[20] |
R. Showalter, Monotone operators in banach spaces and nonlinear partial differential equations, AMS, Providence, 1997.
doi: 10.1090/surv/049. |
[21] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata Serie, 148 (1987), 65-96.
doi: 10.1007/BF01762360. |
[22] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[23] |
R. Triggian and P. F. Yao,
Carleman estimates with no lower-order terms for general riemann wave equations. global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375.
doi: 10.1007/s00245-002-0751-5. |
[24] |
P. F. Yao,
On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.
doi: 10.1137/S0363012997331482. |
[25] |
P. F. Yao,
Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61 (2010), 191-233.
doi: 10.1007/s00245-009-9088-7. |
[26] |
P. F. Yao,
Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differ. Equ., 241 (2007), 62-93.
doi: 10.1016/j.jde.2007.06.014. |
[27] |
P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics. A differential geometric approach. Chapman Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.
doi: 10.1201/b11042.![]() ![]() ![]() |
[28] |
Z. F. Zhang and P. F. Yao,
Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks, SIAM J. Control Optim., 47 (2008), 2044-2077.
doi: 10.1137/070679454. |
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