Existence of random attractors for strongly damped wave equations with multiplicative noise unbounded domain

Year 2021, Volume: 50 Issue: 2, 492 - 510, 11.04.2021

Abdelmajid Issaq , Qiaozhen Ma Ahmed Eshag Mohamed

https://doi.org/10.15672/hujms.614217

https://izlik.org/JA33HD44YJ

Abstract

In this paper, we establish the existence of a random attractor for a random dynamical system generated by the non-autonomous wave equation with strong damping and multiplicative noise when the nonlinear term satisfies a critical growth condition.

Keywords

stochastic strong damped wave equation , asymptotic behavior of solutions , random attractor , random dynamical system

References

• [1] A.D. Abdelmajid, E.M. Ahmed and Q. Ma, Random attractors for Stochastic strongly damped non-autonomous wave equations with memory and multiplicative noise, Open J. Math. Anal. 3, 50–70, 2019.
• [2] L. Arnold and H. Crauel, Random dynamical systems lyapunov exponents proceedings oberwolfach, in: Lecture Notes in Mathematics, 1486, 1–22, 1990.
• [3] P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction diffusion equations on unbounded domains, J. Differ. Equ. 246, 845–869, 2009.
• [4] T. Caraballo and J. Langa, On the upper semi-continuity of cocycle attractors for non-autonomous and random dynamical systems, Discr. Contin. Dyn. Impuls. Syst. Ser. Math. Anal. 10 491–513, 2003.
• [5] T. Caraballo, J. Langa and J. Robinson, Upper semi-continuity of attractors for small random perturbations of dynamical systems, Commun. Part. Differ. Equ. 23, 1557– 1581, 1998.
• [6] F. Chen, B. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differ. Equ. 147, 231–241,1998.
• [7] J.W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. 64 174–187, 2006.
• [8] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002.
• [9] H. Crauel, A. Debussche, and F. Flandoli, Random attractors, J. Dyn. Differ. Equ. 9, 307–341, 1997.
• [10] H. Crauel and F. Fladli Crauel, Attractors for a random dynamical systems, Probab. Theory Related Fields 100, 365–393, 1994.
• [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
• [12] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differ- ential equations, Ann. Probab. 31, 2109–2135, 2003.
• [13] M.M. Fadlallah, A.D. Abdelmajid, E.M. Ahmed, M.Y.A. Bakhet and Q. Ma, Random attractors for semilinear reaction diffusion equation with distribution derivatives and multiplicative noise, Open J. Math. Sci. 4, 126–141, 2020.
• [14] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math. 19, 421-437, 2008.
• [15] F. Flandoli and B. Schmalfuss, Random attractors for the 3-D stochastic Navier- Stokes equation with multiplicative noise, Stoch. Rep. 59, 21–45, 1996.
• [16] J.M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equa- tions, global attractors and their dimension, SIAM J. Math. Anal. 22, 879–895, 1991.
• [17] J.K. Hale and G. Raugel, Upper semi-continuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equ. 73 (2), 197–214, 1988.
• [18] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly- damped wave equation, J. Differ. Equ. 247, 1120–1155, 2009.
• [19] H. Li, Y. You and J. Tu, Random attractors and averaging for nonautonomous sto- chastic wave equations with nonlinear damping, J. Differ. Equ. 258, 148–190, 2015.
• [20] H. Li and S. Zhou, One-dimensional global attractor for strongly damped wave equa- tions, Commun. Nonlinear Sci. Numer. Simul. 12, 784–793, 2007.
• [21] H. Li and S. Zhou, On non-autonomous strongly damped wave equations with a uni- form attractor and Some averaging, J. Math. Anal. Appl. 341, 791–802, 2008.
• [22] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Equ. 48, 334–349, 1983.
• [23] H. Morimoto, Attractors of probability measures for semi-linear stochastic evolution equations, Stoch. Anal. Appl. 10, 205–212, 1992.
• [24] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlin- earity, 19, 1495–1506, 2006.
• [25] A. Pazy, Semigroup of linear operators and applications to partial differential equa- tions, Appl. Math. Sci., Springer, New York, 1983.
• [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
• [27] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Part. Differ. Equ. 253, 1544–1583, 2012.
• [28] B.Wang, Random attractors for non-autonomous stochastic wave equations with mul- tiplicative noise, Discrete Contin. Dyn. Syst. 34, 269–300, 2014.
• [29] B. Wang, Existence and upper semi-continuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn. 14 (4), 1450009, 2014.
• [30] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Spec. 2009, 800–809, 2009.
• [31] Z. Wang and S. Zhou, Asymptotic behavior of stochastic strongly wave equation on unbounded domains, J. Appl. Math. Phys. 3, 338-357, 2015.
• [32] M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc. 361, 1069– 1101, 2009.
• [33] M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear. Anal: Real World Appl. 11, 913–919, 2010.
• [34] F. Yin and L. Liu, D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, J. Comput. Math. Appl. 68, 424–438, 2014.
• [35] S. Zhou and X. Fan, Kernel sections for non-autonomous strongly damped wave equa- tions, J. Math. Anal. Appl. 275, 850–869, 2002.