On a Mixed Problem for a Constant Coefficient Second-Order System. (15th December 2010)

• Record Type: Journal Article
• Authors: Cavazzoni Cavazzoni, Rita Rita
• Other Names: Bates Bates Peter W. Peter W. Academic Editor.
• Is Part Of: Boundary value problems. Volume 2010(2010)
• Publication Date: 2010-12-15
• Subjects: Boundary value problems -- Periodicals; Boundary value problems; Electronic journals; Periodicals; 515.35
• DOI: 10.1155/2010/526917
• Languages: English
• ISSNs: 1687-2762

Abstract

Abstract : The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip Ω A = R d − 1 × ( 0, A ), where A is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in Ω A and the time-variable belongs to the bounded interval ] 0, T [, with T sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution u ∈ L 2 ( [ 0, T ] × [ 0, A ], H 3 ( R d − 1 ) ), by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.