On uniform quasiasymptotics of solutions of the second mixed problem for a~hyperbolic equation

A. K. Gushchin ; V. P. Mikhailov

Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 409-427

Résumé

This paper is devoted to the study of uniform quasiasymptotics of the solution of the second mixed problem in $(0,+\infty)\times\Omega$, $\Omega\in\mathbf R_n$, and of the Cauchy problem $(\Omega=\mathbf R_n)$ for the linear hyperbolic equation $$ u_{tt}-\sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}=f(t,x) $$ with initial conditions $$ u|_{t=0}=\varphi(x),\qquad u_t|_{t=0}=\psi(x). $$ A criterion for the existence of quasiasymptotics of the solution of order $\alpha+2$ is established under the assumption that the function $F(t,x)=f(t,x)\theta(t)+\psi(x)\delta(t)+\varphi(x)\delta'(t)$ has quasiasymptotics of order $\alpha$ and with a certain condition of “isoperimetric type” on the class of domains $\Omega$ considered. Bibliography: 13 titles.