Propagation of a solution of a hyperbolic equation with rapidly oscillating coefficients
A L Pyatnitskiĭ · Russian Mathematical Surveys · 1981
Here the coefficients eyO», τ) are smooth and periodic in R +1 and satisfy the condition of uniform ellipticity. We omit the summation sign for repeated indices. We also assume that the lowest coefficients are smooth and bounded on any compact set. Finally, suppose that the initial functions φ(χ) and ψ(χ) have compact support. In this note we consider the behaviour of the support of a solution of (1) for small e. We denote by Q the domain of dependence [1 ] of (1) with vertex at the point (XQ, 0), and by ^XO(T) the section of this set by the plane {t = τ). Theorem 1. There is a convex cone Qx~ with rectilinear generators, whose form does not depend on x0, such that for all Τ > 0