The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays*

Affiliation(s)

Department of Mathematics, Yunnan University, Kunming, China.

School of Mathematics and Science, Kaili University, Kaili, China.

Department of Mathematics, Yunnan University, Kunming, China.

School of Mathematics and Science, Kaili University, Kaili, China.

ABSTRACT

In this paper, we study the global and pullback attractors for a strongly damped wave equation with delays when the force term belongs to different space. The results following from the solution generate a compact set.

Cite this paper

G. Lin, F. Xia and G. Xu, "The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays*,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 4, 2013, pp. 209-218. doi: 10.4236/ijmnta.2013.24029.

G. Lin, F. Xia and G. Xu, "The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays*,"

References

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http://dx.doi.org/10.1007/978-94-010-0732-0

[2] R. Temam, “Infinite Dimensional Dynamical Systems in Mechanics and Physics,” Springer-Verlag, New York, 1988. http://dx.doi.org/10.1007/978-1-4684-0313-8

[3] C. K. Zhong, M. H. Yang and C. Y. Sun, “The Existence of Global Attractors for the Norm-to-Weak Continuous Semigroup,” Journal of Differential Equations, Vol. 223, No. 2, 2006, pp. 367-399.

[4] Y. Q. Xie and C. k. Zhong, “The Existence of Global Attractor for a Class of Nonlinear Evolution Equation,” Journal of Applied Analysis, Vol. 336, No. 1, 2007, pp. 54-69. http://dx.doi.org/10.1016/j.jmaa.2006.03.086

[5] T. Caraballo, P. E. Kloeden and J. Real, “Pullback and Forward Attractor for a Damped Wave Equation with Delays,” Stochastics and Dynamics, Vol. 4, No. 3, 2004, pp. 405-423.

http://dx.doi.org/10.1142/S0219493704001139

[6] V. Pata and M. Squassina, “On the Strongly Damped Wave Equation,” Communications in Mathematical Physics, Vol. 253, No. 3, 2005, pp. 511-533.

http://dx.doi.org/10.1007/s00220-004-1233-1

[7] T. Caraballo and J. A. Langa, “On the Upper Semicontinuity of Cocycle Attractors for Non-Autonomous and Random Dynamical Systems,” Dynamics of Continuous Discrete and Impulsive Systems, Vol. 10, 2003, pp. 491-513.

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http://dx.doi.org/10.1016/j.jde.2004.04.012

[10] D. Cheban, P. E. Kloeden and B. Schmalfuss, “The Relationship between Pullback, Forwards and Global Attractors of Nonautonomous Dynamical Systems,” Nonlinear Dynamics and Systems Theory, Vol. 2, 2002, pp. 9-28.

[11] C. Y. Sun, S. h. Wang and C. K. Zhong, “Global Attractors for a Nonlassical Diffusion Equation,” Acta Mathematica Sinica, English Series, Vol. 26B, No. 3, 2005, pp. 1-8.

[12] J. Hale, “Asymptotic Behavior of Dissipative Systems,” American Mathematical Society, Providence, 1988.

[13] M. J. Garrido-Atienza and J. Real, “Existence and Uniqueness of Solutions for Delay Evolution Equations of Second Order in Time,” Journal of Mathematical Analysis and Applications, Vol. 283, No. 2, 2003, pp. 582-609.

[14] M. H. Yang, J. Q. Duan and P. Kloeden, “Asymptotic Behavior of Solutions for Random Wave Equations with Nonliear Damping and White Noise,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 1, 2011, pp. 464-478. http://dx.doi.org/10.1016/j.nonrwa.2010.06.032

[15] T. Caraballo, G. Lukaszewicz and J. Real, “Pullback Attractors for Non-Autonomous 2D-Navier-Stokes Equations in Some Unbounded Domains,” Comptes Rendus de l’Académie des Sciences, Vol. 342, No. 4, 2006, pp. 263-268.

[16] Y. J. Wang, C. K. Zhong and S. F. Zhou, “Pullback Attractors for Non-Autonomous Dynamical Systems,” Discrete and Continuous Dynamical Systems, Vol. 16, 2006, pp. 587-614.

[17] Z. J. Yang, “Global Attractor for a Nonlinear Wave Equation Arising in Elastic Waveguide Model,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 5, 2009, pp. 2132-2142.

http://dx.doi.org/10.1016/j.na.2008.02.114

[18] Z. J. Yan and X. Li, “Finite-Dimensional Attractors for the Kirchhoff Equation with a Strong Dissipation,” Journal of Mathematical Analysis and Applications, Vol. 375, No. 2, 2011, pp. 579-593.

http://dx.doi.org/10.1016/j.jmaa.2010.09.051

[19] Y. Q. Xie and C. K. Zhong, “Asymptotic Behavior of a Class of Nonlinear Evolution Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 11, 2009, pp. 5095-5105.

http://dx.doi.org/10.1016/j.na.2009.03.086

[20] S. B. Wang and G. W. Chen, “Cauchy Problem of the Generalized Double Dispersion Equation,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 64, No. 1, 2006, pp. 159-173.

http://dx.doi.org/10.1016/j.na.2005.06.017

[1] J. C. Robinson, “Infinite Dimensional Dynamical Systems,” Cambridge University Press, London, 2001.

http://dx.doi.org/10.1007/978-94-010-0732-0

[2] R. Temam, “Infinite Dimensional Dynamical Systems in Mechanics and Physics,” Springer-Verlag, New York, 1988. http://dx.doi.org/10.1007/978-1-4684-0313-8

[3] C. K. Zhong, M. H. Yang and C. Y. Sun, “The Existence of Global Attractors for the Norm-to-Weak Continuous Semigroup,” Journal of Differential Equations, Vol. 223, No. 2, 2006, pp. 367-399.

[4] Y. Q. Xie and C. k. Zhong, “The Existence of Global Attractor for a Class of Nonlinear Evolution Equation,” Journal of Applied Analysis, Vol. 336, No. 1, 2007, pp. 54-69. http://dx.doi.org/10.1016/j.jmaa.2006.03.086

[5] T. Caraballo, P. E. Kloeden and J. Real, “Pullback and Forward Attractor for a Damped Wave Equation with Delays,” Stochastics and Dynamics, Vol. 4, No. 3, 2004, pp. 405-423.

http://dx.doi.org/10.1142/S0219493704001139

[6] V. Pata and M. Squassina, “On the Strongly Damped Wave Equation,” Communications in Mathematical Physics, Vol. 253, No. 3, 2005, pp. 511-533.

http://dx.doi.org/10.1007/s00220-004-1233-1

[7] T. Caraballo and J. A. Langa, “On the Upper Semicontinuity of Cocycle Attractors for Non-Autonomous and Random Dynamical Systems,” Dynamics of Continuous Discrete and Impulsive Systems, Vol. 10, 2003, pp. 491-513.

[8] T. Caraballo, P. Marin-Rubio and J. Valero, “Autonomous and Non-Autonomous Attractors for Differential Equations with Delays,” Journal of Differential Equations, Vol. 208, No. 1, 2005, pp. 9-41.

[9] T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes Models with Delays,” Journal of Differential Equations, Vol. 205, No. 2, 2004, pp. 270-296.

http://dx.doi.org/10.1016/j.jde.2004.04.012

[10] D. Cheban, P. E. Kloeden and B. Schmalfuss, “The Relationship between Pullback, Forwards and Global Attractors of Nonautonomous Dynamical Systems,” Nonlinear Dynamics and Systems Theory, Vol. 2, 2002, pp. 9-28.

[11] C. Y. Sun, S. h. Wang and C. K. Zhong, “Global Attractors for a Nonlassical Diffusion Equation,” Acta Mathematica Sinica, English Series, Vol. 26B, No. 3, 2005, pp. 1-8.

[12] J. Hale, “Asymptotic Behavior of Dissipative Systems,” American Mathematical Society, Providence, 1988.

[13] M. J. Garrido-Atienza and J. Real, “Existence and Uniqueness of Solutions for Delay Evolution Equations of Second Order in Time,” Journal of Mathematical Analysis and Applications, Vol. 283, No. 2, 2003, pp. 582-609.

[14] M. H. Yang, J. Q. Duan and P. Kloeden, “Asymptotic Behavior of Solutions for Random Wave Equations with Nonliear Damping and White Noise,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 1, 2011, pp. 464-478. http://dx.doi.org/10.1016/j.nonrwa.2010.06.032

[15] T. Caraballo, G. Lukaszewicz and J. Real, “Pullback Attractors for Non-Autonomous 2D-Navier-Stokes Equations in Some Unbounded Domains,” Comptes Rendus de l’Académie des Sciences, Vol. 342, No. 4, 2006, pp. 263-268.

[16] Y. J. Wang, C. K. Zhong and S. F. Zhou, “Pullback Attractors for Non-Autonomous Dynamical Systems,” Discrete and Continuous Dynamical Systems, Vol. 16, 2006, pp. 587-614.

[17] Z. J. Yang, “Global Attractor for a Nonlinear Wave Equation Arising in Elastic Waveguide Model,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 5, 2009, pp. 2132-2142.

http://dx.doi.org/10.1016/j.na.2008.02.114

[18] Z. J. Yan and X. Li, “Finite-Dimensional Attractors for the Kirchhoff Equation with a Strong Dissipation,” Journal of Mathematical Analysis and Applications, Vol. 375, No. 2, 2011, pp. 579-593.

http://dx.doi.org/10.1016/j.jmaa.2010.09.051

[19] Y. Q. Xie and C. K. Zhong, “Asymptotic Behavior of a Class of Nonlinear Evolution Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 11, 2009, pp. 5095-5105.

http://dx.doi.org/10.1016/j.na.2009.03.086

[20] S. B. Wang and G. W. Chen, “Cauchy Problem of the Generalized Double Dispersion Equation,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 64, No. 1, 2006, pp. 159-173.

http://dx.doi.org/10.1016/j.na.2005.06.017