A complete upper estimate on the localization for the degenerate parabolic equation with nonlinear source. (7th February 2014)

• Record Type: Journal Article
• Authors: Zheng, Pan; Mu, Chunlai
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 4(2015:Mar. 15)
• Page Start: 630
• Page End: 635
• Publication Date: 2014-02-07
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3094
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma3094-abs-0001"> <title>Abstract</title> <p id="mma3094-para-0001">This paper deals with the Cauchy problem for the degenerate parabolic equation with a strongly nonlinear source <disp-formula content-type="mathematics" id="mma3094-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3jjvkjz5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma3094:mma3094-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>|</mml:mo><mml:mo>∇</mml:mo><mml:mi>u</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mo>, </mml:mo><mml:mtext> </mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>N</mml:mi></mml:msup><mml:mo>×</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula></p> <p id="mma3094-para-0002">where <italic>N</italic> ≥ 1, <italic> p</italic> > 2, <italic> q</italic> ≥ <italic>p</italic> − 1, and the blow‐up time <italic>T</italic> < ∞ . It has been shown that the solution <italic>u</italic>(<italic>x</italic>, <italic>t</italic>) is strictly localized for <italic>q</italic> ≥ <italic>p</italic> − 1, provided that the initial function <italic>u</italic><sub>0</sub>(<italic>x</italic>) has a compact support by Liang and Zhao. In addition, if <italic>q</italic> > 2<italic>p</italic> − 1, an upper estimate on the localization in terms of the initial support and the blow‐up time <italic>T</italic> is partially derived by Liang. In this work, by using the De Giorgi‐type iteration technique, we give a complete estimate on the localization for all <italic>q</italic> ≥ <italic>p</italic> − 1. Copyright © 2014 John Wiley & Sons, Ltd.</p> </abstract>