Critical exponents for the heat-conduction equation with nonlocal, nonlinear perturbations
type: Item
instance of: scholarly article
title: Critical exponents for the heat-conduction equation with nonlocal, nonlinear perturbations
zbMATH Open document ID: 1090.35030
author: G. I. Laptev
published in: Journal of Mathematical Sciences (New York)
publication date: 2006-01-23T00:00:00Z
review text: The author deals with the existence and nonexistence of a nonnegative solutions (globally defined in time \(t> 0\)) of the problem \[ u_t(x,t)- \Delta_x u(x,t)= u^p(x,t) \Biggl(\int_{\mathbb{R}^N} u^q(y, t)\,dy\Biggr),\quad x\in \mathbb{R}^N,\quad t> 0,\tag{1} \] \[ u(x,0)= u_0(x)\geq 0,\quad x\in\mathbb{R}^N,\quad p> 1,\quad q> 0,\tag{2} \] where \(u(x,t)\) is an unknown function, \(N\geq 1\), \(\Delta_x\) is the Laplace operator with respect to the \(x\in\mathbb{R}^N\). The author provides conditions on \(p\) and \(q\) that guarantee the existence or nonexistence of global in \(t\), nonnegative solutions for (1)-(2) which do not vanish identically.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5000629
zbMATH Keywords: nonnegative solutions
zbMATH Keywords: critical exponent
zbMATH Keywords: existence and nonexistence theorems
MaRDI profile type: Publication
full work available at URL: s10958-005-0300-2
OpenAlex ID: W1987569184
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DOI: 10.1007/S10958-005-0300-2
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description: scientific article; zbMATH DE number 5000629