ON THE ASYMPTOTICS OF SOLUTIONS TO THE NEUMANN PROBLEM FOR HYPERBOLIC SYSTEMS IN DOMAINS WITH CONICAL POINTS

Hyperbolic systems of second-order differential equations are considered in a domain with conical points at the boundary; in particular, the equations of elastodynamics are discussed. The asymptotics of solutions near conical points is studied. The "hyperbolic character" of the asymptotics shows itself in the properties of the coefficients (stress intensity factors) depending on time. Some formulas for the coefficients are presented and sharp estimates in Soboloev's norms are proved. Let G be a domain in R n with boundary ∂G containing conical points. We consider a class of hyperbolic systems of second-order differential equations with Neumann's bound- ary conditions in the cylinder G × R = {(x, t ): x ∈ G, t ∈ R} (and in the semicylinder G × R+). In particular, this class includes the dynamical equations of elasticity the- ory. Our main purpose is to study the asymptotics of solutions near the conical points. We investigate the solvability of the problem mentioned above in a scale of weighted spaces. This enables us to obtain and justify asymptotic formulas. For the coefficients in the asymptotics (depending on time), we give explicit formulas and sharp estimates in Sobolev's norms. The principal part of the asymptotics near a conical point is a linear combination � cj(t)uj(x) of functions uj satisfying a homogeneous elliptic problem in the "tangent" cone; the latter problem is the elliptic part of the initial problem. The hyperbolic char- acter of the asymptotics shows itself in the coefficients cj. They admit representations of the form