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A.</a>; <a href="/authors/milla-miranda.manuel" title="Author Profile">Milla Miranda, M.</a></div> <h2 class="title"> <strong>On a nonlinear wave equation with damping.</strong> <i>(English)</i> <a class="label nowrap" href="/0721.35044">Zbl 0721.35044</a> </h2> <div class="source"> <a href="/serials/1257" title="Journal Profile">Rev. Mat. Univ. Complutense Madr.</a> <a href="/?q=in%3A221251" title="Articles in this Issue">3, No. 2-3, 213-231 (1990)</a>. </div> <div class="abstract">The authors are concerned with global existence and exponential decay for solutions of the mixed problem: \[ \frac{\partial^ 2u}{\partial t^ 2}-M(\int_{\Omega}| \nabla u(x,t)|^ 2dx)\Delta u+(- \Delta)^{\alpha}\frac{\partial u}{\partial t}=f,\text{ in } Q,\text{ with } 0<\alpha \leq 1\quad (*) \]<br class="zbmathjax-paragraph">\[ u(x,t)=0\text{ for } (x,t)\in \Sigma;\quad u(x,0)=u_ 0(x),\quad \frac{\partial u}{\partial t}(x,0)=u_ 1(x)\text{ in } \Omega, \] where M(s) is a positive continuous function on \([0,\infty [;\Omega\) is a bounded open set of \(R^ n\), with smooth boundary \(\Gamma\) ; Q is the cylinder \(\Omega\times]0,\infty [\) of \(R^{n+1}\), with lateral boundary \(\Sigma =\Gamma \times [0,\infty [\); \(\Delta =\sum^{n}_{i=1}\partial^ 2/\partial x_ i^ 2\) is the Laplace operator and \(| \nabla u(x,t)|^ 2=\sum^{n}_{i=1}| \partial u/\partial x_ i|^ 2.\) <br class="zbmathjax-paragraph">They obtain global solutions of the mixed problem (1), that is, mixed problem for (*) with the damping \((-\Delta)^{\alpha}\partial u/\partial t\), if \(0<\alpha \leq 1\), when \(u_ 0\in H^ 1_ 0(\Omega)\cap D((- \Delta)^{\alpha})\), \(u_ 1\in L^ 2(\Omega)\), \(f\in L^ 1(0,T;L^ 2(\Omega))\). And supposing \(M\in C^ 1([0,\infty [;R)\) they prove uniqueness for 1/2\(\leq \alpha \leq 1\).<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A404">Y.Ebihara (Fukuoka)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A4188338">Cited in <strong>27</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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><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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+wave+equation">nonlinear wave equation</a>; <a href="/?q=ut%3Aglobal+existence">global existence</a>; <a href="/?q=ut%3Aexponential+decay">exponential decay</a>; <a href="/?q=ut%3Adamping">damping</a>; <a href="/?q=ut%3Auniqueness">uniqueness</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0721.35044" data-ciurl="/ci/04188338" data-biburl="/bibtex/04188338.bib" data-amsurl="/amsrefs/04188338.bib" data-xmlurl="/xml/04188338.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04188338.pdf" title="Zbl 0721.35044 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://eudml.org/doc/43662" title="Open Access at EuDML">EuDML</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 = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 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