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Mat. Zametki, 1978, Volume 23, Issue 3, Pages 389–400 (Mi mz8154)  

A mixed boundary-value problem for a hyperbolic-parabolic equation

I. E. Egorov

Novosibirsk State University

Abstract: Let $\Omega$ be a bounded domain in the $n$-dimensional Euclidean space. In the cylindrical domain $Q_T=\Omega\times[0,T]$ we consider a hyperbolic-parabolic equation of the form
$$ Lu=k(x,t)u_{tt}+\sum_{i=1}^na_iu_{tx_i}-\sum_{i,j=1}^n\frac\partial{\partial x_i}(a_{ij}(x,t)u_{x_j})+\sum^n_{i=1}b_iu_{x_i}+au_t+cu=f(x,t),\eqno(1) $$
where $k(x,t)\ge0$, $a_{ij}=a_{ji}$, $\nu|\xi|^2\le a_{ij}\xi_i\xi_j\le\mu|\xi|^2$, $\forall \xi\in\mathbf R^n$, $\nu>0$.
The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces $W_2^1(Q_T)$ и $W_2^2(Q_T)$.

Full text: PDF file (686 kB)

English version:
Mathematical Notes, 1978, 23:3, 211–217

Bibliographic databases:

UDC: 517.9
Received: 27.01.1976

Citation: I. E. Egorov, “A mixed boundary-value problem for a hyperbolic-parabolic equation”, Mat. Zametki, 23:3 (1978), 389–400; Math. Notes, 23:3 (1978), 211–217

Citation in format AMSBIB
\Bibitem{Ego78}
\by I.~E.~Egorov
\paper A~mixed boundary-value problem for a~hyperbolic-parabolic equation
\jour Mat. Zametki
\yr 1978
\vol 23
\issue 3
\pages 389--400
\mathnet{http://mi.mathnet.ru/mz8154}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=492923}
\zmath{https://zbmath.org/?q=an:0404.35076|0387.35051}
\transl
\jour Math. Notes
\yr 1978
\vol 23
\issue 3
\pages 211--217
\crossref{https://doi.org/10.1007/BF01651434}


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