A multi-dimensional quenching problem due to a concentrated nonlinear source in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: A multi-dimensional quenching problem due to a concentrated nonlinear source in \(\mathbb R^N\)
zbMATH Open document ID: 1176.35011
author: B. E. Eshmatov
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-08-27T00:00:00Z
review text: The authors consider the Cauchy problem for the equation \(u_t-\Delta u=\alpha\delta_b(|x|)f(u)\) in \(\mathbb R^N\times(0,T]\), where \(\alpha>0\) and \(\delta_b\) is the one-dimensional Delta distribution at \(b>0\). Moreover \(f\), \(f'\) and \(f''\) are positive in the interval \([0,c)\) and \(\lim_{u\to c^-}f(u)=\infty\). Under suitable assumptions they show that a vanishing initial datum evolves into a nonnegative solution which quenches, that is reaches height \(c\), in finite time and exactly on the sphere with radius \(b\). Some related results are also given.
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5316883
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: Delta distribution
MaRDI profile type: Publication
full work available at URL: j.na.2007.07.001
OpenAlex ID: W1555048762
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DOI: 10.1016/J.NA.2007.07.001
description: scientific article; zbMATH DE number 5316883