No Access

Pullback attractors for a strongly damped delay wave equation in $ℝ^{n}$

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P. R. China

E-mail Address: wangyj@lzu.edu.cn

Search for more papers by this author

Yizhao Qin

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P. R. China

Search for more papers by this author

, and

Jingyu Wang

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0219493718500168Cited by:4 (Source: Crossref)

Abstract

In this paper, we prove the existence of a pullback attractor for a strongly damped delay wave equation in $ℝ^{n}$ . We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions, so that some new methods to obtain the existence of pullback attractors for multi-valued processes on an unbounded domain are introduced.

Keywords:

AMSC: 34K05, 35B40, 35B41, 35L05

References

1. J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004) 31–52. Crossref, Web of Science, Google Scholar
2. T. Caraballo, P. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn. 4 (2004) 1–19. Link, Google Scholar
3. T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst. 21 (2008) 415–443. Crossref, Web of Science, Google Scholar
4. T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst. 2 (2009) 17–36. Crossref, Web of Science, Google Scholar
5. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics (Amer. Math. Soc., 2002). Google Scholar
6. M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, Vol. 12 (SIAM, 1992). Crossref, Google Scholar
7. F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise, Stoch. Stoch. Rep. 59 (1996) 21–45. Crossref, Google Scholar
8. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer-Verlag, 1993). Crossref, Google Scholar
9. M. He and A. Liu, The oscillation of hyperbolic functional differential equations, Appl. Math. Comput. 142 (2003) 205–224. Web of Science, Google Scholar
10. Z. Y. Hu and Y. J. Wang, Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay, J. Math. Phys. 53 (2012) 072702. Crossref, Web of Science, Google Scholar
11. O. V. Kapustyan, P. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Commun. Pure Appl. Anal. 13 (2014) 1891–1906. Crossref, Web of Science, Google Scholar
12. P. E. Kloeden and B. Schmalfuß, Asymptotical behavior of nonautonomous difference inclusions, Syst. Cont. Lett. 33 (1998) 275–280. Crossref, Web of Science, Google Scholar
13. V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations (Kluwer, 1999). Crossref, Google Scholar
14. I. Kucuk, I. Sadek and Y. Yilmaz, Active control of a smart beam with time delay by Legendre wavelets, Appl. Math. Comput. 218 (2012) 8968–8977. Web of Science, Google Scholar
15. D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of non-autonomous general dynamical systems and differential inclusions, Set-Valued Anal. 16 (2008) 651–671. Crossref, Web of Science, Google Scholar
16. H. Li, Y. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations 258 (2015) 148–190. Crossref, Web of Science, Google Scholar
17. K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations 236 (2007) 460–492. Crossref, Web of Science, Google Scholar
18. P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier–Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst. 26 (2010) 989–1006. Crossref, Web of Science, Google Scholar
19. F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $ℝ^{N}$ with continuous nonlinearity, Asymptotic Anal. 44 (2005) 111–130. Web of Science, Google Scholar
20. S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006) 1561–1585. Crossref, Web of Science, Google Scholar
21. R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, 2nd edn. (Springer-Verlag, 1997). Crossref, Google Scholar
22. B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $R^{3}$ , Trans. Amer. Math. Soc. 363 (2011) 3639–3663. Crossref, Web of Science, Google Scholar
23. J. Wang, F. Meng and S. Liu, Integral average method for oscillation of second order partial differential equations with delays, Appl. Math. Comput. 187 (2007) 815–823. Web of Science, Google Scholar
24. Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations 232 (2007) 573–622. Crossref, Web of Science, Google Scholar
25. Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Appl. Math. 67 (2009) 343–378. Crossref, Web of Science, Google Scholar
26. Y. J. Wang, Pullback attractors of a damped wave equation with delays, Stoch. Dyn. 15 (2015) 1550003. Link, Web of Science, Google Scholar
27. J. Wu, Theory and Applications of Partial Functional Differential Equations (Springer-Verlag, 1996). Crossref, Google Scholar
28. S. F. Zhou, Kernel sections for damped non-autonomous wave equations with linear memory and critical exponent, Quart. Appl. Math. 61 (2003) 731–757. Crossref, Web of Science, Google Scholar