Criterion for a generalized solution in the class \(L_p\) for the wave equation to be in the class \(W_p^1\)
type: Item
instance of: scholarly article
title: Criterion for a generalized solution in the class \(L_p\) for the wave equation to be in the class \(W_p^1\)
zbMATH Open document ID: 1270.35296
DOI: 10.1134/S1064562412060038
author: V. A. Il'in
author: A. A. Kuleshov
published in: Doklady Mathematics
publication date: 2013-04-03T00:00:00Z
review text: This paper concerns generalized solutions to the wave equation \(\partial^2_t u= \partial^2_x u\) for \((x,t)\in Q_T:= [0,l]\times [0,T]\) with zero initial conditions and Dirichlet boundary conditions \(u(0,t)=\mu(t)\), \(u(1,t)= 0\) for \(t\in[0,T]\). The following result is proved: The solution belongs to the Sobolev space \(W^{1,p}(Q_T)\) if and only if the generalized derivative \(\mu'\) exists on \([0,T]\) and \[ \int^T_0 (T-t)|\mu'(t)|\,dt< \infty. \]
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 6149493
zbMATH Keywords: zero initial conditions
zbMATH Keywords: Dirichlet boundary conditions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6149493