On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points

Abstract

In this paper, we deal with the second initial boundary value problem for higher order hyperbolic systems in domains with conical points. We establish several results on the well-posedness and the regularity of solutions.

On Initial-Boundary Value Problems for Hyperbolic Equations in Domains with Conical Points

“…The theorem is proved by repeating almost word for word the proof of Theorem 3.3 of [7]. Here we present the proof of the existence to show that the restriction of negativeness of in that theorem can be omitted.…”

“…The first lemma deals with the regularity of the solution with respect to time variable. It is proved by repeating almost word for word the proof of Theorem 3.4 of [7] with noting that, as in Theorem 2, the assumption ⩾ 0 in [7] can be removed.…”

“…For example, [4,5] the Cauchy-Dirichlet and Cauchy-Neumann problems for second-order hyperbolic systems with the coefficients independent of the time variable were treated in which the asymptotics of the solutions were established with explicit formulas for the coefficients. In [6][7][8][9], initial-boundary value problems for general higher-order hyperbolic equations and systems with the coefficients depending on both spatial and time variable in a domain containing conical points were studied in which the unique solvability, the regularity, and the asymptotic behaviour of the solutions near the conical points were obtained. In the present paper, these results are extended to initial-boundary value problems for general higher-order hyperbolic equations with more general boundary conditions in infinite cylinders with the bases containing conical points.…”