A nonlocal nonlinear boundary value problem for the heat equations
type: Item
instance of: scholarly article
title: A nonlocal nonlinear boundary value problem for the heat equations
zbMATH Open document ID: 0862.35057
author: Jinhai Yan
published in: Chinese Annals of Mathematics. Series B
publication date: 1997-05-26T00:00:00Z
review text: Let \(\Omega\subset\mathbb{R}^n\) (\(n=2\) or 3) be a bounded smooth domain with boundary \(\partial\Omega=\Gamma_1\cup\Gamma_2\), where \(\Gamma_1\cap\Gamma_2=\emptyset\). We consider the following mixed initial-boundary value problem for the heat equation: \[ u'-\Delta u=0\quad\text{in }\Omega\times(0,T), \] \[ u=0\quad\text{in }\Gamma_1\times(0,T),\;u=H\Biggl(\int_{\Gamma_2}{\partial u\over\partial\nu} dS\Biggr)\quad\text{on }\Gamma_2\times(0,T),\;u(0)=u^0\quad\text{on }\Omega\times\{0\}, \] where \(H:{\mathcal D}_H\subset\mathbb{R}\to\mathbb{R}\) is a strictly decreasing surjective function, \(H^{-1}\) grows linearly, and \(\nu\) stands for the outward unit normal vector at the boundary. We use the Galerkin method to prove the following existence and uniqueness result. Theorem. For any fixed initial data \(u^0\in L^2(\Omega)\), there exists a unique weak solution \[ u\in L^2(0,T;V)\cap W^{1,2}(0,T;V')\subset C(0,T;L^2(\Omega)), \] where \(V=\{v\in H^1(\Omega)\mid v|_{\Gamma_1}=0\), \(v|_{\Gamma_2}=\) constant (depending on \(v\))\}, and \(V'\) stands for the dual of \(V\).
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 940907
zbMATH Keywords: mixed boundary conditions
zbMATH Keywords: Galerkin method
zbMATH Keywords: unique weak solution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 940907