In this work, we analyze the asymptotic behavior of solutions to a generalized semilinear damped wave equation with time-dependent coefficients, subject to homogeneous Dirichlet boundary conditions on a smooth bounded domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. The equation involves functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha _\epsilon (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mi>ϵ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta _\epsilon (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>β</mi> <mi>ϵ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _\epsilon (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mi>ϵ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\eta _\epsilon (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>η</mi> <mi>ϵ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> depending continuously on time and on a parameter <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\epsilon \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, and includes a nonlinear term f(u) satisfying suitable growth, regularity, and dissipativity assumptions. By means of a time-rescaling argument, we establish the existence of pullback exponential attractors and pullback attractors whose sections possess uniformly bounded finite fractal dimension. Furthermore, we prove the continuity of the family of pullback exponential attractors and the upper semicontinuity of the family of pullback attractors with respect to the parameter <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>. These results provide a unified framework for analyzing the robustness and regularity of nonautonomous dissipative dynamics under parameter perturbations.

Asymptotic Dynamics and Continuity of Exponential Attractors for a Generalized Nonautonomous Second-Order Equation

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2026年6月19日

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Vinicius T. Azevedo,
Rodiak N. Figueroa-López,
Marcelo J. D. Nascimento

摘要

In this work, we analyze the asymptotic behavior of solutions to a generalized semilinear damped wave equation with time-dependent coefficients, subject to homogeneous Dirichlet boundary conditions on a smooth bounded domain $\Omega \subset \mathbb {R}^N$ $Ω \subset R^{N}$ , $N \ge 3$ $N \geq 3$ . The equation involves functions $\alpha _\epsilon (t)$ $α_{ϵ} (t)$ , $\beta _\epsilon (t)$ $β_{ϵ} (t)$ , $\mu _\epsilon (t)$ $μ_{ϵ} (t)$ , and $\eta _\epsilon (t)$ $η_{ϵ} (t)$ depending continuously on time and on a parameter $\epsilon \in [0,1]$ $ϵ \in [0, 1]$ , and includes a nonlinear term f(u) satisfying suitable growth, regularity, and dissipativity assumptions. By means of a time-rescaling argument, we establish the existence of pullback exponential attractors and pullback attractors whose sections possess uniformly bounded finite fractal dimension. Furthermore, we prove the continuity of the family of pullback exponential attractors and the upper semicontinuity of the family of pullback attractors with respect to the parameter $\epsilon $ $ϵ$ . These results provide a unified framework for analyzing the robustness and regularity of nonautonomous dissipative dynamics under parameter perturbations.