Random dynamics of fractional nonclassical diffusion equations driven by colored noise

Renhai Wang; Yangrong Li; Bixiang Wang

doi:10.3934/dcds.2019165

Article Contents

2019, Volume 39, Issue 7: 4091-4126. Doi: 10.3934/dcds.2019165

This issue Previous Article On the oscillation behavior of solutions to the one-dimensional heat equation Next Article Stability and separation property of radial solutions to semilinear elliptic equations

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

1.
School of Mathematics and statistics, Southwest University, Chongqing 400715, China
2.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: liyr@swu.edu.cn (Yangrong Li)
^* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received: September 30, 2018

Revised: December 31, 2018

Published: April 2019

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Abstract

The random dynamics in $ H^s(\mathbb{R}^n) $ with $ s\in (0,1) $ is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 35B40, 60H15.

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