<p>The aim of this paper is to establish the <Emphasis Type="BoldItalic">limiting behavior</Emphasis> of random attractors for non-autonomous Benjamin-Bona-Mahony equations with <Emphasis Type="BoldItalic">nonlinear</Emphasis> colored noise and time-delay defined on unbounded channels. A suitable condition to control the time-delay term is given and then several necessary uniform estimates on the solutions of the problem are established. The pullback asymptotical compactness of the non-autonomous cocycle associated with non-autonomous BBM equation with nonlinear colored noise and time-delay in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C([-\rho ,0],H_0^1(\mathcal {O}))$</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> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains and weak dissipative structure of the equation. Then the existence of tempered pullback random attractors of the equation is established in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C([-\rho ,0],H_0^1(\mathcal {O}))$</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> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for nonlinear growth diffusion term as well as Lipschitz time-delay term. At last, the upper semi-continuity of two random attractors is investigated by utilizing the strategy of showing that the solution of BBM equation with time-delay convergent to the solution of non-delay BBM equation as the time delay approaches zero. This work extended our previous work [<CitationRef CitationID="CR8">8</CitationRef>, Mathematische Annalen, 2023] to non-autonomous BBM equation with time-delay and further considered the limiting behavior of random attractors.</p>

Limiting Dynamics of BBM Equations with Nonlinear Colored Noise and Time-Delay on Unbounded Channel

期刊论文
发表日期：

2026年3月12日

查看全文

Xuping Zhang,
Pengyu Chen

摘要

The aim of this paper is to establish the limiting behavior of random attractors for non-autonomous Benjamin-Bona-Mahony equations with nonlinear colored noise and time-delay defined on unbounded channels. A suitable condition to control the time-delay term is given and then several necessary uniform estimates on the solutions of the problem are established. The pullback asymptotical compactness of the non-autonomous cocycle associated with non-autonomous BBM equation with nonlinear colored noise and time-delay in $C([-\rho ,0],H_0^1(\mathcal {O}))$ $C ([- ρ, 0], H_{0}^{1} (O))$ is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains and weak dissipative structure of the equation. Then the existence of tempered pullback random attractors of the equation is established in $C([-\rho ,0],H_0^1(\mathcal {O}))$ $C ([- ρ, 0], H_{0}^{1} (O))$ for nonlinear growth diffusion term as well as Lipschitz time-delay term. At last, the upper semi-continuity of two random attractors is investigated by utilizing the strategy of showing that the solution of BBM equation with time-delay convergent to the solution of non-delay BBM equation as the time delay approaches zero. This work extended our previous work [8, Mathematische Annalen, 2023] to non-autonomous BBM equation with time-delay and further considered the limiting behavior of random attractors.