Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 2

Dynamics of stochastic modified Boussinesq approximation equation driven by fractional Brownian motion

Pages: 183 – 209

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n2.a4

Authors

Jianhua Huang (College of Science, National University of Defense Technology, Changsha, China)

Jin Li (College of Science, National University of Defense Technology, Changsha, China)

Tianlong Shen (College of Science, National University of Defense Technology, Changsha, China)

Abstract

The current paper is devoted to stochastic modified Boussinesq approximation equation driven by fractional Brownian motion with $H \in (\frac{1}{4} , \frac{1}{2})$. Based on the different diffusion operators $P \Delta^2$ and $− \Delta$ in stochastic systems, we combine two types operators $\Phi_1 = I$ and a Hilbert-Schmidt operator $\Phi_2 = \Phi$ to guarantee the convergence of the corresponding Wiener-type stochastic integrals, and show the existence and regularity of the stochastic convolution corresponding to the stochastic modified Boussinesq approximation equation. By the Banach modified fixed point theorem in the selected intersection space, the existence and uniqueness of global mild solution are obtained. Finally, the existence of a random attractor for the random dynamical system generated by the mild solution for the modified Boussinesq approximation equation is also established.

Keywords

infinite-dimensional fractional Brownian motion, stochastic modified Boussinesq approximation equation, mild solution, random attractor

2010 Mathematics Subject Classification

35B40, 35Q35, 76D05

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Published 24 June 2014