<p>This paper is concerned with the random dynamics for nonlocal fractional nonclassical diffusion equations with fractional multiplicative noise and time delay defined on unbounded domain. An interesting feature is that the noise has a fractional Laplace operator multiplier, which seems not to appear in any literature for the study of stochastic PDEs. We first show the pullback asymptotic compactness of solutions for the equation with respect to noise intensity and time delay in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1075_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$C([-\rho ,0],H^\alpha (\mathbb {R}^n))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mi>ρ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <msup> <mi>H</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by utilizing the Arzela–Ascoli theorem as well as the methods of spectral decomposition and uniform tail estimates in order to surmount the difficulties caused by the lack of compact Sobolev embeddings on unbounded domains and the weakly dissipative structures of the equation. Then, we prove the existence and uniqueness of pullback random attractors for polynomial growth drift term and Lipschitz time delay term by establishing several uniform estimates of solutions. Finally, we establish the upper semi-continuity of these attractors as either noise intensity or time delay approaches zero.</p>

Limiting behavior of random attractors for nonlocal diffusion equations with fractional Laplace-multiplier noise and time delay

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发表日期：

2025年5月19日

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Pengyu Chen,
Ru Tian,
Xuping Zhang

摘要

This paper is concerned with the random dynamics for nonlocal fractional nonclassical diffusion equations with fractional multiplicative noise and time delay defined on unbounded domain. An interesting feature is that the noise has a fractional Laplace operator multiplier, which seems not to appear in any literature for the study of stochastic PDEs. We first show the pullback asymptotic compactness of solutions for the equation with respect to noise intensity and time delay in $C([-\rho ,0],H^\alpha (\mathbb {R}^n))$ $C ([- ρ, 0], H^{α} (R^{n}))$ by utilizing the Arzela–Ascoli theorem as well as the methods of spectral decomposition and uniform tail estimates in order to surmount the difficulties caused by the lack of compact Sobolev embeddings on unbounded domains and the weakly dissipative structures of the equation. Then, we prove the existence and uniqueness of pullback random attractors for polynomial growth drift term and Lipschitz time delay term by establishing several uniform estimates of solutions. Finally, we establish the upper semi-continuity of these attractors as either noise intensity or time delay approaches zero.