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doi: 10.3934/cpaa.2021043
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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

* Corresponding author

Received  October 2020 Revised  January 2021 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.

Citation: Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] J. Ghidaglia and R. Temam, Attractors of damped nonlinear hyperbolic equations, J. Math. Pure Appl., 66 (1987), 273-319.   Google Scholar [12] J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988. doi: 10.1090/surv/025.  Google Scholar [13] A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, In MathematicalReports, Harwood Gordon Breach, NewYork, 1987.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] R. Showalter, Monotone operators in banach spaces and nonlinear partial differential equations, AMS, Providence, 1997. doi: 10.1090/surv/049.  Google Scholar [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.  Google Scholar [22] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] J. Ghidaglia and R. Temam, Attractors of damped nonlinear hyperbolic equations, J. Math. Pure Appl., 66 (1987), 273-319.   Google Scholar [12] J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988. doi: 10.1090/surv/025.  Google Scholar [13] A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, In MathematicalReports, Harwood Gordon Breach, NewYork, 1987.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] R. Showalter, Monotone operators in banach spaces and nonlinear partial differential equations, AMS, Providence, 1997. doi: 10.1090/surv/049.  Google Scholar [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.  Google Scholar [22] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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