Classes of uniqueness and solvability of mixed problems for certain evolution equations in unbounded domains
type: Item
instance of: scholarly article
title: Classes of uniqueness and solvability of mixed problems for certain evolution equations in unbounded domains
zbMATH Open document ID: 0773.35012
DOI: 10.1007/BF00971185
author: I. P. Sleptsova
author: A. E. Shishkov
published in: Siberian Mathematical Journal
publication date: 1992-09-26T00:00:00Z
review text: The authors consider the mixed problem \[ (1) \quad u|_{t=0}=u_ t|_{t=0}=0, \qquad (2) \quad D_ x^ \alpha u|_{\partial \Omega\times(0,T)}=0 \quad (|\alpha|\leq m) \] for the linear equation \[ u_{tt}+Au_ t+Bu=F(x,t) \tag{3} \] in a cylindrical domain \(\Omega\times(0,T)\), where \(\Omega\) is a domain in \(\mathbb{R}^ n\), \(0<T<\infty\), \[ A=\sum_{|\alpha|, |\beta| \leq m} (- 1)^{|\beta|} D^ \beta (a_{\alpha\beta} D^ \alpha) \qquad\text{and}\qquad B=\sum_{|\alpha|, |\beta| \leq m+1} (-1)^{| \beta|} D^ \beta (b_{\alpha\beta} D^ \alpha) \] are uniformly elliptic operators with non-smooth coefficients \(a_{\alpha\beta}\) and \(b_{\alpha\beta}\). A priori estimates like Saint-Venant's principle for an arbitrary solution of the corresponding homogeneous problem are obtained, and on the basis of these estimates under some growth restrictions for the function \(F(x,t)\) the existence of a generalized solution of the problem (1)--(3) is an unbounded domain is proved.
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 56513
MaRDI profile type: Publication
cites work: Q3231524
cites work: Q4069114
cites work: Q3667281
cites work: Q3808544
cites work: Q3913792
cites work: Q5548525
cites work: Q3760959
full work available at URL: https://doi.org/10.1007/bf00971185
OpenAlex ID: W2041756106
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description: scientific article; zbMATH DE number 56513