Comparison theorems for solutions of hyperbolic equations

A. K. Gushchin ; V. P. Mikhailov

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 349-371

Résumé

This paper is devoted to the study of uniform quasiasymptotics of the solution of the second mixed problem for the uniformly hyperbolic equation \begin{equation} \begin{gathered} p(x)u_{tt}-\sum^n_{i,j=1}(a_{ij}(x)u_{x_i})_{x_j}=f(t,x),\qquad t>0,\quad x\in\Omega, \\ \frac{\partial u}{\partial N} \biggl|_{\partial\Omega}=0,\quad u|_{t=0}=\varphi(x),\quad u_t|_{t=0}=\psi(x), \end{gathered} \end{equation} where $\Omega$ is an unbounded domain in $\mathbf R_n$ which satisfies certain conditions, the main one of which is a condition of “isoperimetric” type, and $N$ is the conormal to $\partial\Omega$. One of the results is a comparison theorem in which necessary and sufficient conditions are established for the existence of uniform quasiasymptotics of the solution of problem (1) if the uniform quasiasymptotics is known to exist for the solution of a problem differing from problem (1) only by the coefficient of the second derivative with respect to time. Bibliography: 22 titles.