Document Zbl 1522.35075

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Math.</a> <a href="/?q=in%3A501724" title="Articles in this Issue">43, No. 5, 689-701 (2023)</a>. </div> <div class="abstract">Summary: In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem \[ u_{tt}-M\left(\int_\Omega|\nabla u|^2 dx\right)\triangle u-\delta\triangle u_t=\mu|u|^{\rho-2}u\quad \text{in }\Omega\times]0,\infty[, \] where \[ M(s)=\begin{cases}a-bs &\text{for }s\in[0,\frac{a}{b}[,\\ 0, &\text{for }s\in[\frac{a}{b},+\infty[.\end{cases} \] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7733856">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B40" title="MSC2020">35B40</a> </td> <td class="space"> Asymptotic behavior of solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L35" title="MSC2020">35L35</a> </td> <td class="space"> Initial-boundary value problems for higher-order hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B33" title="MSC2020">35B33</a> </td> <td class="space"> Critical exponents in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L76" title="MSC2020">35L76</a> </td> <td class="space"> Higher-order semilinear hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L80" title="MSC2020">35L80</a> </td> <td class="space"> Degenerate hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L70" title="MSC2020">35L70</a> </td> <td class="space"> Second-order nonlinear hyperbolic equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aglobal+solutions">global solutions</a>; <a href="/?q=ut%3Anonlinear+Kirchhoff-type+problem">nonlinear Kirchhoff-type problem</a>; <a href="/?q=ut%3Aexponential+decay">exponential decay</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1522.35075" data-ciurl="/ci/07733856" data-biburl="/bibtex/07733856.bib" data-amsurl="/amsrefs/07733856.bib" data-xmlurl="/xml/07733856.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07733856.pdf" title="Zbl 1522.35075 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.7494/OpMath.2023.43.5.689" aria-label="DOI for “Global solutions for a nonlinear Kirchhoff type equation with viscosity”" title="10.7494/OpMath.2023.43.5.689">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">M. 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