On a coupled system of partial differential equations
type: Item
instance of: scholarly article
title: On a coupled system of partial differential equations
zbMATH Open document ID: 0632.35013
DOI: 10.4171/ZAA/236
author: Gottfried Bruckner
author: Reinhard Kluge
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1987-01-01T00:00:00Z
review text: We investigate the following coupled system of stationary partial differential equations: (1) \[ \nabla \cdot (D_ i\nabla u_ i)+\nabla \cdot (k_ iz_ iu_ i(a+bu^ 2_ 0)^{-}\nabla \psi)=f_ i;\quad -\nabla \cdot (D_ 0\nabla \psi)=k(N+u_ 0),\quad i=1,...,M \] where \(u_ i\), \(f_ i\), \(\psi\) and N are real functions on a bounded domain G of the n-dimensional Euclidean space \({\mathbb{R}}_ n\) allowing partial integration and the Sobolev imbedding theorems. \(\nabla \{D\nabla u)\) means div(D grad u). Let further \(u_ 0=z_ 1u_ 1+...+z_ Mu_ M\), \(z_ i=\pm 1\), \(U_ 0=1/(a+bu^ 2_ 0)^{1/2}\), M a natural number, k, \(k_ i\), a, b positive constants. The diffusion coefficients may be of the form \(D_ i=D_ i(| \nabla u_ i|)\) and \(D_ 0=D_ 0(| \nabla \psi |)\). For all functions \(u_ i\) and \(\psi\) we suppose the same type of boundary conditions: (2) \[ u_ i|_{\partial G_ 1}=\underline y_ i,\quad i=1,...,M,\quad \psi |_{\partial G_ 1}=\underline y_ 0;\quad J_ i\cdot \vec n|_{\partial G_ 2}=0;\quad i=0,1,...,M \] where \(J_ 0=-D_ 0\nabla \psi\) and \(J_ i=-D_ i\nabla u_ i-k_ iz_ iu_ i(a+bu_ 0^ 2)^{-}\nabla \psi\) are the flow densities, \(\partial G_ 1\) and \(\partial G_ 2\) are portions of the boundary of G with the properties \(\partial G_ 1\cap \partial G_ 2=\emptyset\), \(\partial G=\partial G_ 1\cup \partial G_ 2\), mes \(\partial G_ 1>0\) with respect to the boundary measure, and \(\vec n\) is outer normal to the boundary of G. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.
Mathematics Subject Classification ID: 35G30
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4027976
zbMATH Keywords: coupled system
zbMATH Keywords: stationary partial differential equations
zbMATH Keywords: diffusion coefficients
zbMATH Keywords: existence
zbMATH Keywords: energetic method
zbMATH Keywords: upper estimate
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4027976