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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