Document Zbl 1165.34032

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Differ. Equations</a> <a href="/?q=in%3A235412" title="Articles in this Issue">246, No. 1, 428-451 (2009)</a>. </div> <div class="abstract">The one-dimensional Poisson-Nernst-Planck (PNP) system is a basic model for ion flow through membrane channels. If the Debye length is much smaller than the characteristic radius of the channel, the PNP system can be treated as a singularly perturbed system. The paper provides a geometric framework for the study of the steady-state PNP system involving multiple types of ion species with multiple regions of piecewise constant permanent charge.<br class="zbmathjax-paragraph">The article starts with a description of a three-dimensional PNP system as the model for ion flows through an ion channel and discusses a one-dimensional reduction. Special structures of this particular problem are revealed, which together with the general framework allow one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations. Near each singular orbit, an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of a solution of the singular boundary value problem. A new phenomenon on multiplicity and spatial behavior of steady-state involving three or more types of ion species is discovered in an example.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7433">Vasile Dragan (Bucureşti)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5486338">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5486338">Cited in <strong>62</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34E15" title="MSC2020">34E15</a> </td> <td class="space"> Singular perturbations for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34E20" title="MSC2020">34E20</a> </td> <td class="space"> Singular perturbations, turning point theory, WKB methods for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A26" title="MSC2020">34A26</a> </td> <td class="space"> Geometric methods in ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92C35" title="MSC2020">92C35</a> </td> <td class="space"> Physiological flow </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34B15" title="MSC2020">34B15</a> </td> <td class="space"> Nonlinear boundary value problems for ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AGeometric+singular+perturbation">Geometric singular perturbation</a>; <a href="/?q=ut%3Amultiple+steady-states">multiple steady-states</a>; <a href="/?q=ut%3Aspatial+oscillations">spatial oscillations</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 1165.34032" data-ciurl="/ci/05486338" data-biburl="/bibtex/05486338.bib" data-amsurl="/amsrefs/05486338.bib" data-xmlurl="/xml/05486338.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05486338.pdf" title="Zbl 1165.34032 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.2008.09.010" aria-label="DOI for “One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species”" title="10.1016/j.jde.2008.09.010">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">N. 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