Document Zbl 1428.35232

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In honor of Prof. Fernández-Cara’s 60th birthday. Based on talks given at the workshop, Sevilla, Spain, January 25–27, 2017. Cham: Springer. SEMA SIMAI Springer Ser. 17, 17-32 (2018). </div> <div class="abstract">Summary: We consider the initial boundary value problem for nonlinear damped wave equations of the form $u''+M(\int_\Omega|(-\Delta)^su|^2 dx)\Delta u+(-\Delta)^\alpha u'=f$, with Neumann boundary conditions. We prove global existence of solutions, when $s\in [1/2,1]$ and $\alpha\in(0,1]$, and we show that the energy of these ones decays exponentially, as $t\to\infty$. The uniqueness of solutions is also obtained when $\alpha\in[1/2,1]$.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1406.35006">Zbl 1406.35006</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7065842">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L72" title="MSC2020">35L72</a> </td> <td class="space"> Second-order quasilinear hyperbolic equations </td> </tr><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%3A74K10" title="MSC2020">74K10</a> </td> <td class="space"> Rods (beams, columns, shafts, arches, rings, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L20" title="MSC2020">35L20</a> </td> <td class="space"> Initial-boundary value problems for second-order hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R09" title="MSC2020">35R09</a> </td> <td class="space"> Integro-partial differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adamped+wave+equations">damped wave equations</a>; <a href="/?q=ut%3ANeumann+boundary+conditions">Neumann boundary conditions</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 1428.35232" data-ciurl="/ci/07065842" data-biburl="/bibtex/07065842.bib" data-amsurl="/amsrefs/07065842.bib" data-xmlurl="/xml/07065842.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07065842.pdf" title="Zbl 1428.35232 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/10.1007/978-3-319-97613-6_2" aria-label="DOI for “Well-posedness and asymptotic behavior for a nonlinear wave equation”" title="10.1007/978-3-319-97613-6_2">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2024 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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