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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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[3]

J. M. Ball, Attractors of damped wave equations,, Conference at Oberwolfach (Germany), 10 (1992), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

[7]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. V. Coffman, A mininum principle for a class of nonlinear integral equations,, J. Analyse Math., 22 (1969), 391.  doi: 10.1007/BF02786802.  Google Scholar

[11]

E. Feireisl, Finite dimensional asymptotic behavior of some semilinear damped hyperbolic problems,, J. Dynam. Differential Equations, 6 (1994), 23.  doi: 10.1007/BF02219186.  Google Scholar

[12]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419.  doi: 10.1017/S0017089506003156.  Google Scholar

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[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.  Google Scholar

[24]

M. Struwe, Variational Methods,, Springer-Verlage Berlin Heidelberg, (2000).   Google Scholar

[25]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997).   Google Scholar

[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.  Google Scholar

[27]

C. K. Zhong, B. You and R. Yang, The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems,, (preprint)., ().   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[3]

J. M. Ball, Attractors of damped wave equations,, Conference at Oberwolfach (Germany), 10 (1992), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

[7]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. V. Coffman, A mininum principle for a class of nonlinear integral equations,, J. Analyse Math., 22 (1969), 391.  doi: 10.1007/BF02786802.  Google Scholar

[11]

E. Feireisl, Finite dimensional asymptotic behavior of some semilinear damped hyperbolic problems,, J. Dynam. Differential Equations, 6 (1994), 23.  doi: 10.1007/BF02219186.  Google Scholar

[12]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419.  doi: 10.1017/S0017089506003156.  Google Scholar

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[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.  Google Scholar

[24]

M. Struwe, Variational Methods,, Springer-Verlage Berlin Heidelberg, (2000).   Google Scholar

[25]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997).   Google Scholar

[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.  Google Scholar

[27]

C. K. Zhong, B. You and R. Yang, The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems,, (preprint)., ().   Google Scholar

[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.  Google Scholar

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