Dynamics of non-autonomous stochastic dissipative Hamiltonian amplitude wave equations with rapidly oscillating force

ABSTRACT

This paper considers the limiting behavior of pullback attractors for the non-autonomous stochastic dissipative Hamiltonian amplitude wave equation with rapidly oscillating force $g^{\mathit{ϵ}}(x,t)$ defined on a real line. The presence of the complex valued term $i{u}_x$ brings some difficulties even to prove the induced system is pullback absorbing. Thus, by choosing a new number $\mathit{\delta }>0$, some new uniform asymptotic estimates are used to prove the existence of pullback attractors in $H^1(\mathbb{R})\times L^2(\mathbb{R})$, as well as to show the upper semicontinuity of the family of oscillating attractors when $g^{\mathit{ϵ}}(x,t)$ tends to the average $g^0(x,t)$ as the density of noise $\mathit{ϵ}$ approaches to zero.

KEYWORDS:

• Hamiltonian amplitude equation
• rapidly oscillating force
• upper semicontinuity
• random attractor
• unbounded domain

AMS SUBJECT CLASSIFICATIONS:

• 35B40
• 37L55
• 60H15

Acknowledgements

The authors would like to thank the referees for carefully reading the manuscript and valuable suggestions which lead to an improvement of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [11571283]; Postgraduate Research and Innovation Project of Chongqing [CYB2015057]; Natural Science Foundation of Education of Guizhou [KY[2016]103].