Document Zbl 07955155

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China, Math.</a> <a href="/?q=in%3A521308" title="Articles in this Issue">67, No. 12, 2713-2752 (2024)</a>. </div> <div class="abstract">Summary: In this paper, for the 3D quadratic nonlinear Klein-Gordon equation on the product space $\mathbb{R}^2 \times \mathbb{T}$, we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data. When the size of initial data is bounded by $\epsilon_0 >0$, it is shown that a smooth solution exists up to the time $\mathrm{e}^{c_0}/\epsilon_0^2$ with $\epsilon_0$ being sufficiently small and $c_0 > 0$ being some suitable constant. Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on $\mathbb{R}^2 \times \mathbb{T}$ only admits the optimal time-decay rate $(1+t)^{-1}$, from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to $\mathrm{e}^{c_0 /\epsilon_0}$ rather than the more precise $\mathrm{e}^{c_0 /\epsilon_0^2}$ here.</div> <div class="clear"></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%3A35B44" title="MSC2020">35B44</a> </td> <td class="space"> Blow-up in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L15" title="MSC2020">35L15</a> </td> <td class="space"> Initial value problems for second-order hyperbolic equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aquasilinear+Klein-Gordon+equation">quasilinear Klein-Gordon equation</a>; <a href="/?q=ut%3Aalmost+global+solution">almost global solution</a>; <a href="/?q=ut%3A%5C%28Z%5C%29-norm">$Z$-norm</a>; <a href="/?q=ut%3ALittlewood-Paley+decomposition">Littlewood-Paley decomposition</a>; <a href="/?q=ut%3Aspace-time+resonance+method">space-time resonance method</a>; <a href="/?q=ut%3Aenergy+estimate">energy estimate</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 07955155" data-ciurl="/ci/07955155" data-biburl="/bibtex/07955155.bib" data-amsurl="/amsrefs/07955155.bib" data-xmlurl="/xml/07955155.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07955155.pdf" title="Zbl 07955155 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/s11425-023-2219-1" aria-label="DOI for “Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space $\mathbb{R}^2 \times \mathbb{T}$”" title="10.1007/s11425-023-2219-1">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2204.08130"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bambusi, D., Birkhoff normal form for some nonlinear PDEs, Comm Math Phys, 234, 253-285, 2003 · <a href="/1032.37051" class="nowrap">Zbl 1032.37051</a> · <a href="https://doi.org/10.1007/s00220-002-0774-4" class="nowrap">doi:10.1007/s00220-002-0774-4</a></td> </tr><tr> <td>[2]</td> <td class="space">Bambusi, D.; Grébert, B., Birkhoff normal form for partial differential equations with tame modulus, Duke Math J, 135, 507-567, 2006 · <a href="/1110.37057" class="nowrap">Zbl 1110.37057</a> · <a href="https://doi.org/10.1215/S0012-7094-06-13534-2" class="nowrap">doi:10.1215/S0012-7094-06-13534-2</a></td> </tr><tr> <td>[3]</td> <td class="space">Bourgain, J., Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom Funct Anal, 6, 201-230, 1996 · <a href="/0872.35007" class="nowrap">Zbl 0872.35007</a> · <a href="https://doi.org/10.1007/BF02247885" class="nowrap">doi:10.1007/BF02247885</a></td> </tr><tr> <td>[4]</td> <td class="space">Delort, J. 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