Some results on nonlinear heat equations for materials of fading memory type
type: Item
instance of: scholarly article
title: Some results on nonlinear heat equations for materials of fading memory type
zbMATH Open document ID: 0824.45011
author: Guiseppe Da Prato
author: Philippe Clément
published in: Journal of Integral Equations and Applications
publication date: 1990-01-01T00:00:00Z
review text: We consider a model for the heat conduction for a material covering an \(n\)-dimensional bounded set \(\Omega\) with boundary \(\partial\Omega\), \(n= 1,2,3\). \[ {d\over dt} \Biggl(b_ 0 u(t, x)+ \int^ t_ 0 \beta(t- s) u(s, x)ds\Biggr)= c_ 0 \Delta u(t, x),\;t> 0,\;x\in \Omega,\;u(0, x)= x,\;x\in \Omega, \] where \(u(t, x)\) is the temperature of the point \(x\) at time \(t\) (we assume that the temperature is 0 for \(x\in \partial\Omega\)), \(b_ 0\) is the specific heat and \(c_ 0\) the thermal conductivity. We assume that the specific heat has a term of fading memory type \(\int^ t_ 0 \beta(t- s)u(s, x)ds\), whereas the thermal conductivity is constant. Concerning the kernel \(\beta\) we assume only that it is locally integrable in \([0, \infty[\); this will allow us to consider kernels as \(\beta(t)= e^{-\omega t} t^{\alpha- 1}\), \(\omega\geq 0\), \(\alpha\in ]0, 1[\).
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 784745
zbMATH Keywords: nonlinear heat equations
zbMATH Keywords: materials of fading memory type
zbMATH Keywords: heat conduction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075569
OpenAlex ID: W1975036840
DOI: 10.1216/JIEA/1181075569
description: scientific article; zbMATH DE number 784745