<?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%">P. TRAGOONSIRISAK</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">A QUENCHING PROBLEM DUE TO A CONCENTRATED NONLINEAR SOURCE IN AN INFINITE STRIP</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%">35B35</style></keyword><keyword><style  face="normal" font="default" size="100%">35K55</style></keyword><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%">2011</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2011</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://acadsol.eu/dsa/articles/20/33-DSA-31-20.pdf</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">20</style></volume><pages><style face="normal" font="default" size="100%">13</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;This article studies a semilinear parabolic initial-boundary value problem with a concentrated nonlinear source in an infinite strip in the N-dimensional Euclidean space. Existence, uniqueness, and locations where quenching occurs for the solution are investigated.&lt;/p&gt;
</style></abstract><section><style face="normal" font="default" size="100%">505</style></section></record><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%">P. TRAGOONSIRISAK</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">EFFECTS OF A CONCENTRATED NONLINEAR SOURCE ON QUENCHING IN R N</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/18/04-DSA-CY-4-ChanTragoonsirisak.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%">7</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 T and α be positive real numbers, β be a real number, B be a N-dimensional ball  x ∈ R N : |x| &amp;lt; R centered at the origin with a radius R, and ∂B be its boundary. Also, let ν(x) denote the unit inward normal at x ∈ ∂B, and χB(x) be the characteristic function, which is 1 for x ∈ B, and 0 for x ∈ R N \ B. This article studies the following parabolic Cauchy problem with a concentrated nonlinear source on ∂B: ut − 4u = α(1 + |x|) β ∂χB(x) ∂ν f(u) in R N × (0, T], u(x, 0) = 0 for x ∈ R N , u(x, t) → 0 as |x| → ∞ for 0 &amp;lt; t ≤ T, where f is a given function such that limu→c− f(u) = ∞ for some positive constant c, and f(u) and its derivatives f 0 (u) and f 00(u) are positive for 0 ≤ u &amp;lt; c. It is shown that the solution u always quenches for N ≤ 2, and quenching can be prevented for any β for N ≥ 3. For given R and β, the effects of α on quenching are discussed. Similarly for a given α, the effects of R and β on quenching are investigated.&lt;/p&gt;
</style></abstract><section><style face="normal" font="default" size="100%">47</style></section></record><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></authors></contributors><titles><title><style face="normal" font="default" size="100%">QUENCHING CRITERIA FOR A DEGENERATE PARABOLIC PROBLEM DUE TO A CONCENTRATED NONLINEAR SOURCE</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%">35B35</style></keyword><keyword><style  face="normal" font="default" size="100%">35K57</style></keyword><keyword><style  face="normal" font="default" size="100%">35K60</style></keyword><keyword><style  face="normal" font="default" size="100%">35K65</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/18/11-DSA-CY-10-Chan.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%">7</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;A criterion for the quenching of the solution for a degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at b is given. The locations of b for global existence of the solution and for the quenching of the solution are given.&lt;/p&gt;
</style></abstract><section><style face="normal" font="default" size="100%">121</style></section></record><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;
</style></abstract><section><style face="normal" font="default" size="100%">55</style></section></record><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%">H. T. LIU</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">QUENCHING FOR DEGENERATE PARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS</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%">35K20</style></keyword><keyword><style  face="normal" font="default" size="100%">35K55</style></keyword><keyword><style  face="normal" font="default" size="100%">35K57</style></keyword><keyword><style  face="normal" font="default" size="100%">35K60</style></keyword><keyword><style  face="normal" font="default" size="100%">35K65</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/18/02-DSA-CY-2-ChanLiu.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%">12</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 q be a nonnegative real number, and a and T be positive constants. This article studies the following degenerate parabolic problem: x qut − uxx = G(u) in (0, a) × (0, T], where G is a nonnegative function in the form of either f(u(x, t)), or Ra 0 h(x, t) f(u(x, t))dx for some positive, bounded and continuous function h with f &amp;gt; 0, f 0&amp;gt; 0, f 00≥ 0, and limu→1− f(u) = ∞. It is subject to the initial condition, u(x, 0) = 0 on [0, a], and the boundary conditions, u(0, t) = Za 0 M(x)|u (x, t)| p dx, u (a, t) = Za 0 N (x)|u (x, t)| r dx, t &amp;gt; 0, where p and r are constants greater than or equal to 1, and M and N are given nonnegative functions. Existence, uniqueness and criteria for quenching and non-quenching are studied.&lt;/p&gt;
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