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Mat. Sb., 2000, Volume 191, Number 4, Pages 29–52 (Mi msb469)  

This article is cited in 1 scientific paper (total in 1 paper)

Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface

A. V. Deryabina

State Academy of Consumer Services

Abstract: Equations of the following form are considered:
\begin{equation} \psi^2(t,x)u_{tt}+\varphi(t,x)u_t-\sum_{i,j}(a^{ij}(t,x)u_{x_i})_{x_j}+\sum_ib^i(t,x)u_{x_i}+c(t,x)u=f(t,x), \tag{1} \end{equation}
where
\begin{gather*} (t,x)\in H=(0,T)\times\mathbb R^n, \qquad \psi(t,x)\geqslant 0, \qquad \varphi(t,x)\geqslant0;
\sum_{i,j}a^{ij}(t,x)\xi_i\xi_j\geqslant0 \quad \forall (t,x)\in H, \quad \forall \xi=(\xi_1,…,\xi_n)\in\mathbb R^n. \end{gather*}

In place of the Cauchy problem for (1), a problem without initial data but with constraints on the admissible growth of the solution as $t\to0$ and as $|x|\to\infty$ is discussed. The unique solubility of (1) in certain Sobolev-type weighted spaces is proved. The smoothness properties of generalized solutions are studied.

DOI: https://doi.org/10.4213/sm469

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English version:
Sbornik: Mathematics, 2000, 191:4, 503–527

Bibliographic databases:

UDC: 517.956
MSC: Primary 35L80; Secondary 35D10
Received: 12.05.1998 and 17.09.1999

Citation: A. V. Deryabina, “Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface”, Mat. Sb., 191:4 (2000), 29–52; Sb. Math., 191:4 (2000), 503–527

Citation in format AMSBIB
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\by A.~V.~Deryabina
\paper Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 4
\pages 29--52
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\crossref{https://doi.org/10.4213/sm469}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1775041}
\zmath{https://zbmath.org/?q=an:0962.35128}
\transl
\jour Sb. Math.
\yr 2000
\vol 191
\issue 4
\pages 503--527
\crossref{https://doi.org/10.1070/sm2000v191n04ABEH000469}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0034338852}


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    This publication is cited in the following articles:
    1. Zhang K., “The Cauchy Problem For Semilinear Hyperbolic Equation With Characteristic Degeneration on the Initial Hyperplane”, Math. Meth. Appl. Sci., 41:6, SI (2018), 2429–2441  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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