In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x) ∈ L2(Ω). In the framework of time-dependent spaces, we verify the existence of absorbing sets and the asymptotic compactness of the process, then we obtain the existence of the time-dependent global attractor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10255_2024_1036_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr A}={\{A_{t}}\}_{t\in{\mathbb R}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">A</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <msub> <mo fence="false" stretchy="false">}</mo> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> in ℳt. Furthermore, we achieve the regularity of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10255_2024_1036_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">A</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, that is, At is bounded in ℳ<Stack> t 1 </Stack> with a bound independent of t.

Time-dependent Global Attractors for the Nonclassical Diffusion Equations with Fading Memory

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发表日期：

2025年3月1日

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Yu-ming Qin,
Xiao-ling Chen

摘要

In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x) ∈ L²(Ω). In the framework of time-dependent spaces, we verify the existence of absorbing sets and the asymptotic compactness of the process, then we obtain the existence of the time-dependent global attractor ${\mathscr A}={\{A_{t}}\}_{t\in{\mathbb R}}$ $A = {A_{t}}_{t \in R}$ in ℳ_t. Furthermore, we achieve the regularity of ${\mathscr A}$ $A$ , that is, A_t is bounded in ℳ _t ¹ with a bound independent of t.