Localization and blow up of thermal waves in nonlinear heat conduction with peaking
type: Item
instance of: scholarly article
title: Localization and blow up of thermal waves in nonlinear heat conduction with peaking
zbMATH Open document ID: 0627.35048
DOI: 10.1007/BF01456972
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164458
review text: We consider the initial-boundary value problem: \[ u_ t=(u^ m)_{xx}\quad for\quad (x,t)\in (0,\infty)\times (0,t), \] \[ u(x,0)=u_ 0(x)\quad for\quad x\in [0,\infty),\quad u(0,t)=\psi (t)\quad for\quad t\in [0,T), \] where \(m>1\), \(0<T<\infty\), and \(u_ 0\) and \(\psi\) are nonnegative continuous functions satisfying the compatibility condition \(u_ 0(0)=\psi (0)\) and \(\psi\) (t)\(\uparrow \infty\) as \(t\uparrow T\). If \(u_ 0\) is bounded, it is known that this problem has a unique nonnegative generalized solution, u(x,t). Furthermore, if \(u_ 0\) has compact support, the solution possesses a free boundary \(\zeta (t)=\sup \{x\in [0,\infty):\) \(u(x,t)>0\}.\) In the context of nonlinear heat conduction, a solution u(x,t) with such a free boundary is often referred to as a thermal wave. We say that a thermal wave is localized if \(\lim _{t\uparrow T}\sup \zeta (t)<\infty.\) Under the assumption that \(\psi\) is monotonic increasing, we show that localization occurs if and only if \(\lim _{t\uparrow T}\sup [\int ^{t}_{0}\Psi ^ m(s)ds/\Psi (t)]<\infty.\) We also discuss the nature of the blow-up set \(\Omega =\{x\in [0,\infty):\) \(\lim _{t\uparrow T}\sup u(x,t)=\infty \}.\) We show that if localization does not occur, then \(\Omega =[0,\infty)\). On the other hand, if localization does occur, then \(\Omega\) is a bounded interval. In this case, estimates of the size of \(\Omega\) are obtained. These indicate the circumstances under which \(\Omega\) may or may not reduce to the single point \(x=0\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 4019594
zbMATH Keywords: porous media equation
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: nonnegative generalized solution
zbMATH Keywords: free boundary
zbMATH Keywords: nonlinear heat conduction
zbMATH Keywords: thermal wave
zbMATH Keywords: localization
zbMATH Keywords: blow-up
author: Brian H. Gilding
author: Miguel A. Herrero
MaRDI profile type: Publication
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OpenAlex ID: W2071483975
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description: scientific article; zbMATH DE number 4019594