<p>In this paper we consider the well-posedness and asymptotic behavior of a Kirchhoff-type wave equation under the effects of strong damping whose intensity is given by a non-local degenerate coefficient that depends on the energy associated with the linear part of the system. This class of dissipations is connected with energy damping models proposed by Balakrishnan (A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988) and Balakrishnan-Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). In our main result, we establish that for each parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1251_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> associated with the coefficient of the potential part of the Kirchhoff term, the problem has a global attractor <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1251_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {A}}_{\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi>κ</mi> </msub> </math></EquationSource> </InlineEquation>. Moreover, we prove that the family of global attractors <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1251_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\{{\mathcal {A}}_{\kappa }\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="script">A</mi> <mi>κ</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is upper semi-continuous at <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1251_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>.</p>

Dynamics of Kirchhoff Wave Equations Incorporating Energy Damping Effects

期刊论文
发表日期：

2025年3月17日

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Pengyan Ding,
Vando Narciso

摘要

In this paper we consider the well-posedness and asymptotic behavior of a Kirchhoff-type wave equation under the effects of strong damping whose intensity is given by a non-local degenerate coefficient that depends on the energy associated with the linear part of the system. This class of dissipations is connected with energy damping models proposed by Balakrishnan (A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988) and Balakrishnan-Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). In our main result, we establish that for each parameter $\kappa $ $κ$ associated with the coefficient of the potential part of the Kirchhoff term, the problem has a global attractor ${\mathcal {A}}_{\kappa }$ $A_{κ}$ . Moreover, we prove that the family of global attractors $\{{\mathcal {A}}_{\kappa }\}$ ${A_{κ}}$ is upper semi-continuous at $\kappa $ $κ$ .