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January  2014, 19(1): 217-230. doi: 10.3934/dcdsb.2014.19.217

## Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent

 1 Department of Mathematics, Nanjing University, Nanjing, 210093, China 2 Department of Mathematics, Nanjing University, Nanjing 210093

Received  November 2012 Revised  June 2013 Published  December 2013

In this paper, we are concerned with some properties of the global attractor of weakly damped wave equations. We get the existence of multiple stationary solutions for wave equations with weakly damping. Furthermore, we provide some approaches to verify the small neighborhood of the origin is an attracting domain which is important to obtain the multiple equilibrium points in global attractor.
Citation: Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
##### References:
 [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equations with critical exponents,, Comm. Partial Differential Equations, 17 (1992), 841. doi: 10.1080/03605309208820866. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). [3] J. M. Ball, Attractors of damped wave equations,, Conference at Oberwolfach (Germany), 10 (1992), 31. doi: 10.3934/dcds.2004.10.31. [4] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. [5] R. Brown, P. Perry and Z. Shen, On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains,, Indiana Univ. Math. J., 49 (2000), 81. doi: 10.1512/iumj.2000.49.1603. [6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000). doi: 10.1017/CBO9780511526404. [7] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). [8] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [9] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. [10] C. V. Coffman, A mininum principle for a class of nonlinear integral equations,, J. Analyse Math., 22 (1969), 391. doi: 10.1007/BF02786802. [11] E. Feireisl, Finite dimensional asymptotic behavior of some semilinear damped hyperbolic problems,, J. Dynam. Differential Equations, 6 (1994), 23. doi: 10.1007/BF02219186. [12] S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419. doi: 10.1017/S0017089506003156. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). [14] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite- dimensional spaces,, Nonlinearity, 12 (1999), 1263. doi: 10.1088/0951-7715/12/5/303. [15] A. Kh. Khanmamedov, Global attractors for wave equation with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702. doi: 10.1016/j.jde.2006.06.001. [16] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964). [17] I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation,, J. Math. Anal. Appl., 270 (2002), 16. doi: 10.1016/S0022-247X(02)00006-9. [18] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [19] V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611. [20] V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping,, Adv. Math. Sci. Appl., 17 (2007), 225. [21] D. Prazak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping,, J. Dynam. Differential Equations, 14 (2002), 763. [22] G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. [23] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. [24] M. Struwe, Variational Methods,, Springer-Verlage Berlin Heidelberg, (2000). [25] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). [26] C. K. Zhong and W. S. Niu, On the $Z_2$ index of the global attractor for a class of $p$- Laplacian equations,, Nonlinear Anal., 73 (2010), 3698. doi: 10.1016/j.na.2010.07.022. [27] C. K. Zhong, B. You and R. Yang, The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems,, (preprint)., (). [28] S. F. Zhou, Dimension of the global attractor for damped nonlinear wave equation,, Proc. Amer. Math. Soc., 127 (1999), 3623. doi: 10.1090/S0002-9939-99-05121-7.

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##### References:
 [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equations with critical exponents,, Comm. Partial Differential Equations, 17 (1992), 841. doi: 10.1080/03605309208820866. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). [3] J. M. Ball, Attractors of damped wave equations,, Conference at Oberwolfach (Germany), 10 (1992), 31. doi: 10.3934/dcds.2004.10.31. [4] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. [5] R. Brown, P. Perry and Z. Shen, On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains,, Indiana Univ. Math. J., 49 (2000), 81. doi: 10.1512/iumj.2000.49.1603. [6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000). doi: 10.1017/CBO9780511526404. [7] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). [8] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [9] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. [10] C. V. Coffman, A mininum principle for a class of nonlinear integral equations,, J. Analyse Math., 22 (1969), 391. doi: 10.1007/BF02786802. [11] E. Feireisl, Finite dimensional asymptotic behavior of some semilinear damped hyperbolic problems,, J. Dynam. Differential Equations, 6 (1994), 23. doi: 10.1007/BF02219186. [12] S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419. doi: 10.1017/S0017089506003156. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). [14] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite- dimensional spaces,, Nonlinearity, 12 (1999), 1263. doi: 10.1088/0951-7715/12/5/303. [15] A. Kh. Khanmamedov, Global attractors for wave equation with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702. doi: 10.1016/j.jde.2006.06.001. [16] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964). [17] I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation,, J. Math. Anal. Appl., 270 (2002), 16. doi: 10.1016/S0022-247X(02)00006-9. [18] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. [19] V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611. [20] V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping,, Adv. Math. Sci. Appl., 17 (2007), 225. [21] D. Prazak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping,, J. Dynam. Differential Equations, 14 (2002), 763. [22] G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. [23] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. [24] M. Struwe, Variational Methods,, Springer-Verlage Berlin Heidelberg, (2000). [25] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). [26] C. K. Zhong and W. S. Niu, On the $Z_2$ index of the global attractor for a class of $p$- Laplacian equations,, Nonlinear Anal., 73 (2010), 3698. doi: 10.1016/j.na.2010.07.022. [27] C. K. Zhong, B. You and R. Yang, The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems,, (preprint)., (). [28] S. F. Zhou, Dimension of the global attractor for damped nonlinear wave equation,, Proc. Amer. Math. Soc., 127 (1999), 3623. doi: 10.1090/S0002-9939-99-05121-7.
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