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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 $W_2^2(Q_T)$.
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
Linking options:
https://www.mathnet.ru/eng/mzm8154 https://www.mathnet.ru/eng/mzm/v23/i3/p389
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Abstract page: | 233 | Full-text PDF : | 100 | First page: | 1 |
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