<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">C. Y. CHAN</style></author><author><style face="normal" font="default" size="100%">T. TREEYAPRASERT</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">QUENCHING FOR A PARABOLIC PROBLEM DUE TO A CONCENTRATED NONLINEAR SOURCE ON A SEMI-INFINITE INTERVAL</style></title><secondary-title><style face="normal" font="default" size="100%">Dynamic Systems and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">35K57</style></keyword><keyword><style  face="normal" font="default" size="100%">35K60</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2009</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2009</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://acadsol.eu/dsa/articles/18/05-DSA-CY-5-ChanTreeyaprasert.pdf</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">18</style></volume><pages><style face="normal" font="default" size="100%">8</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;Let α, b, and T be positive numbers, D = (0,∞), D¯ = [ 0, ∞ ), and Ω = D × ( 0, T ]. This article studies the first initial-boundary value problem with a concentrated nonlinear source situated at b,&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;u&lt;sub&gt;t&lt;/sub&gt; − u&lt;sub&gt;xx&lt;/sub&gt; = αδ( x − b ) f( u(x, t) )&amp;nbsp; in Ω,&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;u( x, 0 ) = 0 on D¯,&amp;nbsp;&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;u( 0, t ) = 0 and u( x, t ) → 0 as x → ∞&amp;nbsp; for&amp;nbsp; 0 &amp;lt; t ≤ T,&lt;/p&gt;

&lt;p&gt;where δ (x) is the Dirac delta function and f is a given function such that lim&lt;sub&gt;u→c−&lt;/sub&gt; f (u) = ∞ for some positive constant c, and f(u) and its derivatives f´(u) and f''(u) are positive for 0 ≤ u &amp;lt; c. The problem has a unique continuous solution u before sup {u (x, t) : 0 ≤ x &amp;lt; ∞} reaches c&lt;sup&gt;−&lt;/sup&gt;, and u is a strictly increasing function of t in Ω. It is shown that if&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; sup { u (x, t) : 0 ≤ x &amp;lt; ∞ }&lt;/p&gt;

&lt;p&gt;reaches c&lt;sup&gt;−&lt;/sup&gt;, then u attains the value c in a finite time only at the point b. A criterion for u to exist globally and a criterion for u to quench in a finite time are given. It is also shown that there exists a critical position b&lt;sup&gt;∗&lt;/sup&gt; for the nonlinear source to be placed such that for b ≤ b&lt;sup&gt;∗&lt;/sup&gt; , u exists for 0 ≤ t &amp;lt; ∞,&amp;nbsp; and for b &amp;gt; b&lt;sup&gt;∗&lt;/sup&gt; , u quenches in a finite time. This also implies that u does not quench in infinite time. The formula for computing b&lt;sup&gt;∗&lt;/sup&gt; is also derived.&lt;/p&gt;
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