Theorems on the complete set of isomorphisms in the $L_2$-theory of generalized solutions of boundary value problems for a Petrovskii parabolic equation
Abstract:
The general boundary value problem is studied for a parabolic equation in spaces of insufficiently smooth and generalized functions. Starting from Green's formula, the generalized solution of a boundary value problem is defined, and two families (scales) of spaces are constructed in which the boundary value problem is studied: the spaces of solutions $\widetilde{\mathscr H}^s(\Omega)$, and the sapces of right-hand sides $\mathscr K^s(\Omega)$. It is proved that the closure with respect to continuity of the boundary value problem operator establishes an isomorphism of the spaces $\widetilde{\mathscr H}^s(\Omega)$ and $\mathscr K^s(\Omega)$ for $-\infty<s<\infty$.
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