On the critical Fujita exponents for solutions of quasilinear parabolic inequalities
type: Item
instance of: scholarly article
title: On the critical Fujita exponents for solutions of quasilinear parabolic inequalities
zbMATH Open document ID: 1006.35010
DOI: 10.1016/S0022-247X(02)00005-7
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-08T00:00:00Z
review text: This work is devoted to the study of critical blow-up phenomena for the inequality \[ u_t\geq A(u)+|u|^{q-1}- u\tag{1} \] in the half-space \(\mathbb{S}=\mathbb{R}^1_+\times \mathbb{R}^n\), \(n\geq 1\). This problem is considered for wide classes of quasilinear partial differential operators \(A(u)\) in divergent form. Well-known examples of such operators are (for \(\alpha>0)\) \[ A(u)= \text{div}_x\Bigl[ \biggl|\bigl( |u|^{m-1}\cdot u\bigr)_x\biggr |^{ \alpha-2} \cdot\bigl(|u|^{m-1} \cdot u\bigr)_x\Bigr],\;m\geq 1, \] \[ A(u) =\text{div}_x\biggl[ \bigl(|u|^{m-1}\cdot u\bigr)_x\biggr],\;m\geq 1,\;A(u)=\text{div}_x \bigl[|u_x|^{\alpha-2} \cdot u_x\bigr]. \] The authors obtain the existence of critical blow-up exponents for nonnegative weak solutions of the inequality (1) that belong only locally to the corresponding Sobolev spaces in the half-space \(\mathbb{S}\).
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1764671
zbMATH Keywords: blow-up phenomena
author: Athanassios G. Kartsatos
author: V. V. Kurta
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00005-7
OpenAlex ID: W2005287409
description: scientific article; zbMATH DE number 1764671