Document Zbl 1258.35155

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Dyn. Differ. Equations</a> <a href="/?q=in%3A312774" title="Articles in this Issue">24, No. 4, 927-954 (2012)</a>. </div> <div class="abstract">Summary: The current paper is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. It is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. Next we obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems. Finally we discuss the applications of the established principal eigenvalue theory to time periodic Fisher or KPP type equations with nonlocal dispersal and prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution which is globally asymptotically stable.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6132843">Cited in <strong>80</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35P15" title="MSC2020">35P15</a> </td> <td class="space"> Estimates of eigenvalues in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K55" title="MSC2020">35K55</a> </td> <td class="space"> Nonlinear parabolic equations </td> </tr><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%3A45C05" title="MSC2020">45C05</a> </td> <td class="space"> Eigenvalue problems for integral equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlocal+dispersal">nonlocal dispersal</a>; <a href="/?q=ut%3Arandom+dispersal">random dispersal</a>; <a href="/?q=ut%3Aprincipal+eigenvalue">principal eigenvalue</a>; <a href="/?q=ut%3Aprincipal+spectrum+point">principal spectrum point</a>; <a href="/?q=ut%3Avanishing+condition">vanishing condition</a>; <a href="/?q=ut%3Alower+bound">lower bound</a>; <a href="/?q=ut%3Amonostable+equation">monostable equation</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 1258.35155" data-ciurl="/ci/06132843" data-biburl="/bibtex/06132843.bib" data-amsurl="/amsrefs/06132843.bib" data-xmlurl="/xml/06132843.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06132843.pdf" title="Zbl 1258.35155 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/s10884-012-9276-z" aria-label="DOI for “Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications”" title="10.1007/s10884-012-9276-z">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">Apreutesei N., Bessonov N., Volpert V., Vougalter V.: Spatial structures and generalized traveling waves for an integro-differential equation. 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