A general linear 2b-parabolic initial-boundary value problem in a finite cylinder is well-posed on a scale of generalized anisotropic Sobolev (Hörmander) spaces H^{s,s/(2b);φ}, whose regularity is calibrated by a power index s together with a function parameter φ that varies slowly at infinity. The operator sending a solution to (equation right-hand side, boundary data, initial data) is an isomorphism onto the subspace cut out by explicit compatibility conditions at the initial time, obtained by transferring the classical Sobolev-space result across the scale via interpolation with a function parameter between Hilbert spaces. Students learn how the finer φ-calibration yields sharper local-regularity and continuity criteria for generalized derivatives than any power scale can express.
For a single linear partial differential equation that is 2b-parabolic in Petrovskii's sense, posed in a finite cylinder G x (0, tau) with m boundary conditions covering it on the lateral surface and…