<p>In this article we present the existence, asymptotic behavior and numerical analysis for the one-dimensional Signorini’s problem. We prove the existence of at least one global solution of the semi linear model, using the penalized hybrid method introduced in Muñoz Rivera and da Costa Baldez (J Math Anal Appl 458:1274–1291, 2018. <a href="https://doi.org/10.1016/jmaa:2017.10.22">https://doi.org/10.1016/jmaa:2017.10.22</a>). This method is based on approximate the problem through a dynamic boundary condition depending of a parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3083_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> (hybrid model). Using the Lipschitzian perturbation theory, we show the global existence for the semi linear problem. Additionally, we show the existence of solutions to the Signorini’s problem by taking limit <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3083_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this way we prove that the energy associated with the model decays exponentially to zero, as time approaches infinity. Furthermore, we show the exponential decay for the semi-discrete model and that the decay rate does not depend on the penalty parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3083_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, we show the convergence of the approximate model.</p>

Numerical analysis for a dissipative hyperbolic system with a boundary constraints

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发表日期：

2025年1月18日

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C. A. da Costa Baldez,
J. E. Muñoz Rivera

摘要

In this article we present the existence, asymptotic behavior and numerical analysis for the one-dimensional Signorini’s problem. We prove the existence of at least one global solution of the semi linear model, using the penalized hybrid method introduced in Muñoz Rivera and da Costa Baldez (J Math Anal Appl 458:1274–1291, 2018. https://doi.org/10.1016/jmaa:2017.10.22). This method is based on approximate the problem through a dynamic boundary condition depending of a parameter $\epsilon $ $ϵ$ (hybrid model). Using the Lipschitzian perturbation theory, we show the global existence for the semi linear problem. Additionally, we show the existence of solutions to the Signorini’s problem by taking limit $\epsilon \rightarrow 0$ $ϵ \to 0$ . In this way we prove that the energy associated with the model decays exponentially to zero, as time approaches infinity. Furthermore, we show the exponential decay for the semi-discrete model and that the decay rate does not depend on the penalty parameter $\epsilon >0$ $ϵ > 0$ . Finally, we show the convergence of the approximate model.