On the structure of the solutions of the first initial boundary value problem for the Sobolev's equation
type: Item
instance of: scholarly article
title: On the structure of the solutions of the first initial boundary value problem for the Sobolev's equation
zbMATH Open document ID: 0826.35096
published in: Journal of Mathematics of Kyoto University
publication date: 1995-11-23T00:00:00Z
review text: Let \(\Omega\) be a plane convex bounded domain with corner points. The Cauchy-Dirichlet problem is considered for the Sobolev equation \[ {\partial^2 \over \partial t^2} \left( {\partial^2u \over \partial x_1^2} + {\partial^2u \over \partial x^2_2} \right) + {\partial^2u \over \partial x^2_2} = 0, \quad x \in \Omega, \;t > 0. \] This problem is reduced to the abstract Cauchy problem for the equation \(p_{tt} = Ap\), where \(A\) is a bounded selfadjoint operator in \(W_0^{1,2}(\Omega)\) with spectrum \(\sigma (A) = [-1,0]\). Under a certain condition on \(\partial \Omega\) it is shown that \(A\) has no eigenfunctions in \(W_0^{1,2}(\Omega)\) but for every point of \(\sigma (A)\) there exists a class of generalized eigenfunctions. The completeness of these generalized eigenfunctions in \(W_0^{1,2}(\Omega)\) is proved. Integral representation for solutions of the above abstract Cauchy problem is obtained.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 584575
zbMATH Keywords: Sobolev equation
zbMATH Keywords: integral representation
zbMATH Keywords: bounded domain with corner points
zbMATH Keywords: Cauchy-Dirichlet problem
zbMATH Keywords: abstract Cauchy problem
zbMATH Keywords: generalized eigenfunctions
zbMATH Keywords: completeness
author: Andrei A. Lyashenko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519122
OpenAlex ID: W1939169354
DOI: 10.1215/KJM/1250519122
description: scientific article; zbMATH DE number 584575