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The \(L_ 2\)-theory of the generalized solutions of general linear parabolic boundary value problems. (Russian) Zbl 0825.35060

The parabolic boundary value problem \({\mathcal L}u=f\), \(Bu_ s=\varphi\), \(Cu_{t=0}=\psi\), where \({\mathcal L}\), \(B\), and \(C\) are general linear matrix differential operators, is studied in the cylindrical domain \(\Omega= G\times[0,T)\) with the side surface \(S=\partial G\times[0,T)\). The well-posedness of the parabolic boundary value problem in spaces of the generalized functions is indicated. Its solvability in the anisotropic spaces \(\tilde{\mathcal H}^ s(\Omega)\), \(s\in \mathbb{R}^ 1\), of the generalized Sobolev-Slobodetskij functions is proved.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
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