<?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%">ALFONSO C. CASAL</style></author><author><style face="normal" font="default" size="100%">J. ILDEFONSO DIAZ</style></author><author><style face="normal" font="default" size="100%">JOSE M. VEGAS</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">BLOW-UP IN SOME ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS WITH TIME-DELAY</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%">34K05</style></keyword><keyword><style  face="normal" font="default" size="100%">34K12</style></keyword><keyword><style  face="normal" font="default" size="100%">34K40</style></keyword><keyword><style  face="normal" font="default" size="100%">35B05</style></keyword><keyword><style  face="normal" font="default" size="100%">35B30</style></keyword><keyword><style  face="normal" font="default" size="100%">35B40</style></keyword><keyword><style  face="normal" font="default" size="100%">35B60</style></keyword><keyword><style  face="normal" font="default" size="100%">35B65</style></keyword><keyword><style  face="normal" font="default" size="100%">35K55</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/18/03-DSA-CY-3-Casal.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%">18</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;Blow-up phenomena are analyzed for both the delay-differential equation (DDE) u ′ (t) = B ′ (t)u(t − τ), and the associated parabolic PDE (PDDE) ∂tu = ∆u + B ′ (t)u(t − τ, x), where B : [0, τ] → R is a positive L 1 function which behaves like 1/ |t − t ∗ | α , for some α ∈ (0, 1) and t ∗ ∈ (0, τ). Here B′ represents its distributional derivative. For initial functions satisfying u(t ∗ − τ) &amp;gt; 0, blow up takes place as t ր t ∗ and the behavior of the solution near t ∗ is given by u(t) ≃ B(t)u(t − τ), and a similar result holds for the PDDE. The extension to some nonlinear equations is also studied: we use the Alekseev’s formula (case of nonlinear (DDE)) and comparison arguments (case of nonlinear (PDDE)). The existence of solutions in some generalized sense, beyond t = t ∗ is also addressed. This results is connected with a similar question raised by A. Friedman and J. B. McLeod in 1985 for the case of semilinear parabolic equations.&lt;/p&gt;
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