Finite fractal dimension of random attractors for non-autonomous fractional stochastic reaction–diffusion equations in ℝ

Abstract

This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction–diffusion equations with multiplicative noise in $\mathbb{R}$. We prove the existence and uniqueness of the tempered pullback random attractor for the equations in $L^2\left(\mathbb{R}\right)$ and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional Laplacian operator is non-local and the Sobolev embedding is not compact on unbounded domains. To solve this, we derive the tail-estimates of solutions of the equation and decompose the solutions into a sum of three parts, which one part is finite-dimensional and other two parts are quickly decay in mean sense.

Keywords:

• Random dynamical system
• fractional reaction–diffusion equation
• multiplicative noise
• random attractor
• fractal dimension

2010 Mathematics Subject Classifications:

• 60H15
• 37L55
• 35Q56

Acknowledgments

The authors would like to thank the reviewers for their helpful comments. This work was partially supported by the National Natural Science Foundation of China (No.11871138), the funding of V.C. & V.R. Key Lab of Sichuan Province.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [No. 11871138], the funding of V.C. & V.R. Key Lab of Sichuan Province.