On the limit behavior of the domain of dependence of a~hyperbolic equation with rapidly oscillating coefficients

A. L. Piatnitski

Sbornik. Mathematics, Tome 43 (1982) no. 1, pp. 117-131

Résumé

In this paper, the behavior of the support of the solution to the Cauchy problem for a hyperbolic equation of the form $$ \frac{\partial^2}{\partial t^2}u^\varepsilon(x, t)-\frac\partial{\partial x_i}a_{ij}\biggl(\frac x\varepsilon\biggr)\frac\partial{\partial x_j}u^\varepsilon+b_i\biggl(x, \frac x\varepsilon\biggr)\frac\partial{\partial x_i}u^\varepsilon+c\biggl(x, \frac x\varepsilon\biggr)u^\varepsilon=0 $$ with periodic, rapidly oscillating coefficients $a_{ij}(y)$ and small parameter $\varepsilon$, is studied. It is proved that, for small $\varepsilon$, the domain of dependence of this equation is close to some convex cone with rectilinear generators. In the case when the coefficients $a_{ij}$ depend essentially on only one argument, e.g. $y_1$, this limit cone can be found explicitly. Its construction uses the Hamiltonian, which does not depend on $\varepsilon$ and does not correspond to any differential operator. Bibliography: 8 titles.