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 Mat. Sb. (N.S.), 1971, Volume 85(127), Number 2(6), Pages 189–200 (Mi msb3203)

Some problems for linear partial differential equations with constant coefficients in the entire space and for a class of degenerate equations in a halfspace

A. S. Kalashnikov

Abstract: In the space $\mathbf R^{n+1}=\mathbf R_t^1\times\mathbf R_x^n$ we consider a linear partial differential equation with constant coefficients which is solvable in the leading derivative with respect to $t$. We prove that two problems with limit conditions as $t\to-\infty$ which are imposed on the Fourier transform $F_{x\to\sigma}[u(t,x)]$ and contain weight factors, are uniquely solvable in the class of functions $u(t,x)$ which for every $t$ belong to $L_2(\mathbf R_x^n)$ along with the derivatives appearing in the equation and which grow at an order no faster that $t$ as $t\to+\infty$ (in $L_2$). We apply these results to a class of equations in a halfspace which degenerate on the boundary hyperplane.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:2, 186–198

Bibliographic databases:

UDC: 517.947
MSC: 35G15, 35J70

Citation: A. S. Kalashnikov, “Some problems for linear partial differential equations with constant coefficients in the entire space and for a class of degenerate equations in a halfspace”, Mat. Sb. (N.S.), 85(127):2(6) (1971), 189–200; Math. USSR-Sb., 14:2 (1971), 186–198

Citation in format AMSBIB
\Bibitem{Kal71} \by A.~S.~Kalashnikov \paper Some problems for linear partial differential equations with constant coefficients in the entire space and for a~class of degenerate equations in a~halfspace \jour Mat. Sb. (N.S.) \yr 1971 \vol 85(127) \issue 2(6) \pages 189--200 \mathnet{http://mi.mathnet.ru/msb3203} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=284698} \zmath{https://zbmath.org/?q=an:0222.35010} \transl \jour Math. USSR-Sb. \yr 1971 \vol 14 \issue 2 \pages 186--198 \crossref{https://doi.org/10.1070/SM1971v014n02ABEH002612}