Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 231-260

ASYMPTOTICAL DYNAMICS FOR NON-AUTONOMOUS STOCHASTIC EQUATIONS DRIVEN BY A NON-LOCAL INTEGRO-DIFFERENTIAL OPERATOR OF FRACTIONAL TYPE

Wenqiang Zhao

Chongqing Technology and Business University, School of Mathematics and Statistics
Chongqing 400067, P.R. China; gshzhao 'at' sina.com

Abstract. The purpose of this paper is to investigate the asymptotical dynamics of solutions for a non-autonomous stochastic evolution equations driven by a non-local integro-differential operator L_K defined by
L_Ku(x) = P.V. ∫_RN(u(x) – u(y))K(x – y) dy, x ∈ R^N,

where K : R^N \ {0} → (0,+∞) is the kernel of L_K which satisfies the general fractional-type condition of order s. It is showed that the (2p – 2)-truncation of solutions on a finite integral interval vanishes if the initial time goes to negative infinite and the eigenvalue of L_K is large enough. By means of this truncation estimate and the spectrum splitting technique, the flattening condition of solutions is proved in the fractional-type Sobolev space X₀^s, under a weak assumption on the non-autonomous term. Then, the regular dynamics of the cocycle associated with this problem are demonstrated, namely, that the pullback attractor established in L²(O) is actually compact, measurable and attracting in the fractional-type space X₀^s for any s ∈ (0,1) and N > 2s. As a typical example, we derive the random dynamics for the problem driven by the fractional Laplacian (-Δ)^s.

2010 Mathematics Subject Classification: Primary 35R60, 35B40, 35B41, 35B65.

Key words: Non-autonomous integro-differential equation, non-local operator, fractional Laplacian, pullback attractor, additive noise, flattening condition.

Reference to this article: W. Zhao: Asymptotical dynamics for non-autonomous stochastic equations driven by a non-local integro-differential operator of fractional type. Ann. Acad. Sci. Fenn. Math. 44 (2019), 231-260.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4414