On linear hyperbolic boundary-value problems

Abstract

The paper deals with the well posedness of linear hyperbolic second-order systems with homogeneous boundary conditions, in the half-space $\Omega = {\bf R}^{d-1} \times (0,\infty)$. At first we consider operators with constant coefficients: by performing a Fourier-Laplace transform and by studying the space of the solutions, we prove a sufficient condition for the well posedness of the boundary-value problem, by means of the Hille-Yosida theorem. Subsequently, we study the case of evolution boundary-value problems for second-order systems with coefficients that depend on the space variable. Proving that the linear second-order operator associated with the system, under suitable assumptions, turns out to be maximal and monotone, we establish the well posedness of the problem by applying the Hille-Yosida theorem.