Parabolic problems and interpolation with a function parameter. (English) Zbl 1289.35147

Los, Valerii; Murach, Aleksandr A.

Methods Funct. Anal. Topol. 19, No. 2, 146-160 (2013).

In a series of papers, V. A. Mikhailets and the second author developed a version of the theory of elliptic boundary value problems based on the refined Sobolev scale of Hilbert spaces with a functional parameter; see their survey [Banach J. Math. Anal. 6, No. 2, 211–281 (2012; Zbl 1258.46014)].
In the paper under review, this approach is extended to parabolic equations. The authors introduce a refined anisotropic Sobolev scale connected with anisotropic Sobolev spaces by means of interpolation with a functional parameter. The generalized smoothness is characterized by a real number and a function varying slowly at infinity in the sense of Karamata. It is proved that operators corresponding to parabolic initial-boundary value problems determine isomorphisms between appropriate spaces of the new scale.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

35K35	Initial-boundary value problems for higher-order parabolic equations
46B70	Interpolation between normed linear spaces
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

interpolation with functional parameter; parabolic initial-boundary value problem; refined anisotropic Sobolev scale

Citations:

Zbl 1258.46014