Blow-up for a degenerate parabolic equation with a nonlocal source.
type: Item
instance of: scholarly article
title: Blow-up for a degenerate parabolic equation with a nonlocal source.
zbMATH Open document ID: 1049.35114
DOI: 10.1016/S0022-247X(03)00414-1
author: Youpeng Chen
author: Chunhong Xie
author: Qi-lin Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-10-14T00:00:00Z
review text: The authors consider the following non-local degenerate second order parabolic partial differential equation: \[ \begin{aligned} v_t(x, t) &= x^{\alpha} (v^m)_{xx} + \int_0^l v^p(x, t)dx - kv^q(x, t), \quad 0 < x < l,\;t > 0, \\ v(0, t) &= v(l, t) = 0,\quad t > 0,\\ v(x, 0) &= v_0(x), \quad 0 < x < l,\end{aligned} \] where \(0 < \alpha < 2,\) \(p\geq q > m > 1\). Under suitable conditions the existence and uniqueness of solution is established. Also, it is shown that the solution blows up in finite time, and the blow up is global. Moreover, estimates of the blow up rates as well as an example of a suitable initial value \(v_0(x)\), which satisfies the conditions for the blow up, are given.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1992561
zbMATH Keywords: blow-up rate
zbMATH Keywords: local existence and uniqueness
zbMATH Keywords: blow up
reviewed by: Mohamed O. El-Doma
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00414-1
OpenAlex ID: W1994726306
description: scientific article; zbMATH DE number 1992561