On the solvability of parabolic and hyperbolic problems with a boundary integral condition
type: Item
instance of: scholarly article
title: On the solvability of parabolic and hyperbolic problems with a boundary integral condition
zbMATH Open document ID: 1011.35002
DOI: 10.1155/S0161171202005860
author: Abdelfatah Bouziani
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/49981
full work available at URL: S0161171202005860.html
review text: The paper deals with the following two linear equations: \[ (1) \quad v_t + Lv = f, \qquad \quad (2) \quad v_{tt} + Lv = f \] where \(Lv = -( a(x,t)v_x)_x + b(x,t)v\), and the unknown function \(v(x,t)\) is defined for \(\alpha < x < \beta\), \(0 < t < T\). Each of the equations (1), (2) is equipped with initial and boundary conditions of the form \[ v_x(\alpha, t) = \mu (t), \qquad \quad \int_{\alpha}^{\beta} v(x,t) dx = m(t), \qquad \quad 0<t<T. \tag{3} \] Here \(a, b, f, \mu, m\) are known functions and \(a\) is supposed to be bounded away from zero. The author proves a priori estimates for the generalized solution of (1), (3) or (2), (3) in suitably defined function spaces. The above estimates yield the theorem about the existence and uniqueness of the generalized solution.
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 1780417
zbMATH Keywords: one space dimension
zbMATH Keywords: generalized solution
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1780417