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 Mat. Sb. (N.S.), 1986, Volume 131(173), Number 4(12), Pages 419–437 (Mi msb1971)

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

A. K. Gushchin, V. P. Mikhailov

Abstract: 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.

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English version:
Mathematics of the USSR-Sbornik, 1988, 59:2, 409–427

Bibliographic databases:

UDC: 517.9
MSC: 35L15, 35L20, 35B40

Citation: A. K. Gushchin, V. P. Mikhailov, “On uniform quasiasymptotics of solutions of the second mixed problem for a hyperbolic equation”, Mat. Sb. (N.S.), 131(173):4(12) (1986), 419–437; Math. USSR-Sb., 59:2 (1988), 409–427

Citation in format AMSBIB
\Bibitem{GusMik86} \by A.~K.~Gushchin, V.~P.~Mikhailov \paper On uniform quasiasymptotics of solutions of the second mixed problem for a~hyperbolic equation \jour Mat. Sb. (N.S.) \yr 1986 \vol 131(173) \issue 4(12) \pages 419--437 \mathnet{http://mi.mathnet.ru/msb1971} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=881906} \zmath{https://zbmath.org/?q=an:0635.35056} \transl \jour Math. USSR-Sb. \yr 1988 \vol 59 \issue 2 \pages 409--427 \crossref{https://doi.org/10.1070/SM1988v059n02ABEH003144} 

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This publication is cited in the following articles:
1. A. K. Gushchin, V. P. Mikhailov, “Comparison theorems for solutions of hyperbolic equations”, Math. USSR-Sb., 62:2 (1989), 349–371
2. V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111
3. V. Zh. Dumanyan, “On the uniform quasiasymptotics of the solutions of hyperbolic equations”, Math. USSR-Sb., 70:1 (1991), 109–128
4. Denisov V., “On the Stabilization of Mean-Values of Solutions of Cauchy-Problem for 2nd-Order Parabolic Equations”, 315, no. 4, 1990, 777–780
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