<p>We investigate the long-time dynamics of a coupled system of structurally damped wave equations on a smooth bounded domain, with linear symmetric coupling modulated by a parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varkappa > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. As <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varkappa \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, the system exhibits a singular limiting behavior: The two components synchronize and converge to a single strongly damped wave equation with a rescaled nonlinearity. We rigorously justify this singular limit both at the level of trajectories and global attractors. In particular, we establish the convergence of solutions to those of the limiting problem and prove the upper semicontinuity of global attractors with respect to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varkappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϰ</mi> </math></EquationSource> </InlineEquation>, including in the singular regime <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varkappa \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach combines uniform-in-<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varkappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϰ</mi> </math></EquationSource> </InlineEquation> energy bounds, compactness arguments, and quasi-stability techniques, unveiling the structural robustness of the global dynamics under strong coupling limits.</p>

On the singular limit of a coupled structurally damped wave system

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发表日期：

2025年12月19日

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Mirelson M. Freitas,
Jeverson S. Santos

摘要

We investigate the long-time dynamics of a coupled system of structurally damped wave equations on a smooth bounded domain, with linear symmetric coupling modulated by a parameter $\varkappa > 0$ $ϰ > 0$ . As $\varkappa \rightarrow \infty $ $ϰ \to \infty$ , the system exhibits a singular limiting behavior: The two components synchronize and converge to a single strongly damped wave equation with a rescaled nonlinearity. We rigorously justify this singular limit both at the level of trajectories and global attractors. In particular, we establish the convergence of solutions to those of the limiting problem and prove the upper semicontinuity of global attractors with respect to $\varkappa $ $ϰ$ , including in the singular regime $\varkappa \rightarrow \infty $ $ϰ \to \infty$ . Our approach combines uniform-in- $\varkappa $ $ϰ$ energy bounds, compactness arguments, and quasi-stability techniques, unveiling the structural robustness of the global dynamics under strong coupling limits.