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On the uniform quasiasymptotics of solutions of hyperbolic equations. (English. Russian original) Zbl 0754.35080
Math. USSR, Sb. 70, No. 1, 109-128 (1991); translation from Mat. Sb. 181, No. 5, 684-704 (1990).
(Author’s abstract.) The uniform quasiasymptotics as $$t\to\infty$$ of the solutions of the second mixed problem and of the Cauchy problem for a linear hyperbolic second order equation are studied in the scale of self- similar functions. The method of investigation is based on the construction, in terms of a given self-similar function, of a special convolution operator that reduces the study of the quasiasymptotics to that of the power scale discussed earlier [A. K. Gushchin and V. P. Mikhajlov, Mat. Sb., Nov. Ser. 134 (176), No. 3 (11), 353-374 (1987; Zbl 0678.35063); 131 (173), No. 4 (12), 419-437 (1986; Zbl 0635.35056); Dokl. Akad. Nauk SSSR 287, 37-40 (1986; Zbl 0629.35072)].
##### MSC:
 35L15 Initial value problems for second-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
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