Document Zbl 1412.35180

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Differ. Equations</a> <a href="/?q=in%3A432851" title="Articles in this Issue">267, No. 2, 1423-1466 (2019)</a>. </div> <div class="abstract">Summary: The present paper is devoted to the investigation of the following nonlocal dispersal equation \[u_t(t, x) = \frac{D}{\sigma^m} \bigg[\int_{{\Omega}} J_\sigma(x - y) u(t, y) d y - u(t, x)\bigg ] + f(t, x, u(t, x)), t > 0, x \in \overline{{\Omega}},\] where ${\Omega} \subset \mathbb{R}^N$ is a bounded and connected domain with smooth boundary, $m \in [0, 2)$ is the cost parameter, $D > 0$ is the dispersal rate, $\sigma > 0$ characterizes the dispersal range, $J_\sigma = \frac{1}{\sigma^N} J(\frac{\cdot}{\sigma})$ is the scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of generalized KPP type. This paper is a continuation of the works of <span class="zbmathjax-textit">H. Berestycki</span> et al. [J. Funct. Anal. 271, No. 10, 2701–2751 (2016; <a href="/1358.47044">Zbl 1358.47044</a>); J. Math. Biol. 72, No. 7, 1693–1745 (2016; <a href="/1346.35202">Zbl 1346.35202</a>)], where $f$ was assumed to be time-independent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at $u \equiv 0$. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup-inf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate $D$ and the dispersal range characterized by $\sigma$ on the principal spectrum point and the positive time-periodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive time-periodic solution as $D, \sigma \rightarrow 0^+$ or $\infty$. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual $L^2$ variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the time-dependence of the nonlinearity $f$. Finally, we establish the maximum principle for time-periodic nonlocal dispersal operators.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7055546">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7055546">Cited in <strong>29</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K57" title="MSC2020">35K57</a> </td> <td class="space"> Reaction-diffusion equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47G20" title="MSC2020">47G20</a> </td> <td class="space"> Integro-differential operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D25" title="MSC2020">92D25</a> </td> <td class="space"> Population dynamics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37L15" title="MSC2020">37L15</a> </td> <td class="space"> Stability problems for infinite-dimensional dissipative dynamical systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlocal+dispersal+equation">nonlocal dispersal equation</a>; <a href="/?q=ut%3Aprincipal+spectrum+point">principal spectrum point</a>; <a href="/?q=ut%3Ageneralized+principal+eigenvalue">generalized principal eigenvalue</a>; <a href="/?q=ut%3Apositive+solution">positive solution</a>; <a href="/?q=ut%3Aglobal+dynamics">global dynamics</a>; <a href="/?q=ut%3Amaximum+principle">maximum principle</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1358.47044">Zbl 1358.47044</a>; <a href="/1346.35202">Zbl 1346.35202</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 1412.35180" data-ciurl="/ci/07055546" data-biburl="/bibtex/07055546.bib" data-amsurl="/amsrefs/07055546.bib" data-xmlurl="/xml/07055546.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07055546.pdf" title="Zbl 1412.35180 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/10.1016/j.jde.2019.02.013" aria-label="DOI for “Nonlocal dispersal equations in time-periodic media: principal spectral theory, limiting properties and long-time dynamics”" title="10.1016/j.jde.2019.02.013">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1704.00401"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">Andreu, F.; Mazón, J. 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