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June  2022, 21(6): 1857-1871. doi: 10.3934/cpaa.2021043

On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation

Jiacheng Wang 1,2, and  Peng-Fei Yao 1,2,,

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

* Corresponding author

Received  October 2020 Revised  January 2021 Published  June 2022 Early access  March 2021

Fund Project: The work is supported by the National Science Foundation of China, grants no. 12071463 and no. 61573342 and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011

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.

Keywords: attractor, semilinear wave equation, variable coeffi- cients, nonlinear boundary dissipation.
Mathematics Subject Classification: Primary: 93C20, 93D20.
Citation: Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1857-1871. doi: 10.3934/cpaa.2021043
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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