On a system of linear hyperbolic equations defined on a complex

Abstract

WE CONSIDER the system of linear equations Lu = f(L = δ δt + Λ δ δx − A) . (0.1) Here, u and f are vectors with s components, Λ and A are square matrices, and Λ is a matrix of simple structure with real eigenvalues. The system is hyperbolic. If Λ is sufficiently smooth, it can be reduced to the diagonal form [1, 2]. We shall assume from the start that Λ is in fact diagonal. We shall describe as a segment any set of points homeomorphic to the segment of a straight line. A set of segments such that the common point of any two (if they intersect) is the end of each is called a complex [3]. A left- and right-hand end can be distinguished for each segment. Orientation can be introduced into it. For this, we direct the x axis from left to right. A complex with orientation thus introduced on to it can be regarded as a generalization of the x axis. The functions Λ( x, t), A( x, t) and f( x, t) should be defined at each point of the segment (“for each x”) and for each t (0 ⩽ t ⩽ T), in which case system (0.1) has a meaning. The unknown is u. In order for the solution to be unique, boundary conditions need to be specified. These include initial conditions, i.e. the value of u at t = 0, and boundary conditions at the ends of the segments. These can be written briefly as lu = ψ (0.2) The operator l is assumed linear. Its form will be described below. Our main purpose will be to investigate the stability of the solution of system (0.1) under boundary conditions (0.2). We shall call the solution stable in the interval 0 ⩽ t ⩽ T if ∥ u ∥ ⩽ J 1 ∥ ψ ∥ + J 2 ∥ f ∥ ( J 1, J 2 = const). (0.3) The norms of u, ψ and f will be described below. We shall call the solution asymptotically stable if (0.3) holds for 0 ⩽ t ⩽ ∞. The uniqueness of the solution is an obvious consequence of (0.3) and the linearity of the problem.