Parabolic problems with nonlinear dynamical boundary conditions and singular initial data
type: Item
instance of: scholarly article
title: Parabolic problems with nonlinear dynamical boundary conditions and singular initial data
zbMATH Open document ID: 1018.35047
author: Pavol Quittner
author: Jose M. Arrieta
author: Anibal Rodriguez-Bernal
published in: Differential and Integral Equations
publication date: 2003-03-13T00:00:00Z
review text: This paper is devoted to the following parabolic problem with dynamic boundary conditions: \[ \begin{cases} u_t+{\mathcal A}u=f(x,t,u,\nabla u),\quad & x \in \Omega,\;t>0,\\ (\gamma u)_t+ \beta u=g(x,t, \gamma u),\quad & x\in \Gamma,\;t>0.\\ u(x,0)= u_0(x),\quad & x\in\Omega,\\ (\gamma u)(x,0)=v_0(x),\quad & x\in \Gamma, \end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) of class \(C^2\), \(\Gamma=\partial \Omega\), \(\gamma\) is the trace operator, and \({\mathcal A}u= -\Delta u+\omega u\), \(\beta u=u_\nu+\omega u\), \(\nu\) denotes the outer normal on \(T\). Here the authors are interested in obtaining the largest possible growth on the nonlinear terms such that problem (1) can be solved with initial data in Lebesgue spaces, \(L_{q_1}(\Omega)\times L_{q_2} (\Gamma)\), or even measures \({\mathcal M}(\Omega) \times{\mathcal M}(\Gamma)\), where \({\mathcal M}(\Omega)\) and \({\mathcal M}(\Gamma)\) denote the space of bounded Radon measures in \(\Omega\) and on \(\Gamma\), respectively.
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1832879
zbMATH Keywords: measure initial data
zbMATH Keywords: largest possible growth
zbMATH Keywords: bounded Radon measures
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1832879