Abstract:
This article is devoted to a study of the behaviour of the solution to the Cauchy problem for the quasihyperbolic equation (1) (defined below in §1).
For such equations, as we shall show, certain regions inside the base of the characteristic cone can turn out to be lacunae or weak lacunae (defined in §1).
Next we show that each quasihyperbolic equation (1) can be regarded as the limit for some hyperbolic equation whose coefficients in the series of higher derivatives in t tend to zero. We establish a connection between fundamental solutions to the Cauchy problem for both equations.
The statements of the main results have been published in [1].
\Bibitem{Gal74}
\by S.~A.~Gal'pern
\paper Fundamental solutions and lacunae of quasihyperbolic equations
\jour Russian Math. Surveys
\yr 1974
\vol 29
\issue 2
\pages 158--169
\mathnet{http://mi.mathnet.ru/eng/rm4360}
\crossref{https://doi.org/10.1070/RM1974v029n02ABEH003840}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=415051}
\zmath{https://zbmath.org/?q=an:0294.35049|0305.35064}
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https://www.mathnet.ru/eng/rm4360
https://doi.org/10.1070/RM1974v029n02ABEH003840
https://www.mathnet.ru/eng/rm/v29/i2/p154
This publication is cited in the following 6 articles:
V. I. Korzyuk, Ya. V. Rudko, “Klassicheskoe reshenie zadachi Koshi dlya polulineinogo giperbolicheskogo uravneniya v sluchae dvukh nezavisimykh peremennykh”, Izv. vuzov. Matem., 2024, no. 3, 50–63
V. I. Korzyuk, J. V. Rudzko, “Classical Solution to the Cauchy Problem for a Semilinear Hyperbolic Equation in the Case of Two Independent Variables”, Russ Math., 68:3 (2024), 41
C.H. Daros, “A fundamental solutions for transversely isotropic, piezoelectric solids under electrically irrotational approximation”, Mechanics Research Communications, 29:1 (2002), 61
A. A. Lokshin, “Rasprostranenie vozmuschenii ot tochechnogo istochnika,
opisyvaemoe kvazigiperbolicheskim uravneniem”, UMN, 31:5(191) (1976), 243–244
B. R. Vainberg, “On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t→∞ of solutions of non-stationary problems”, Russian Math. Surveys, 30:2 (1975), 1–58
A. A. Lokshin, “Usloviya suschestvovaniya slabykh lakun”, UMN, 30:3(183) (1975), 165–166
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