A complete upper estimate on the localization for the degenerate parabolic equation with nonlinear source. (7th February 2014) - Zheng, Pan; Mu, Chunlai
Abstract This paper deals with the Cauchy problem for the degenerate parabolic equation with a strongly nonlinear source <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3jjvkjz5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ut=div(|∇u|p−2∇u)+uq, (x, t)∈ℝN×(0, T), </alternatives> where N ≥ 1, p > 2, q ≥ p − 1, and the blow‐up time T < ∞ . It has been shown that the solution u(x, t) is strictly localized for q ≥ p − 1, provided that the initial function u0(x) has a compact support by Liang and Zhao. In addition, if q > 2p − 1, an upper estimate on the localization in terms of the initial support and the blow‐up time T 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 q ≥ p − 1. Copyright © 2014 John Wiley & Sons, Ltd.