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# Parabolic problems and interpolation with a function parameter

### Abstract

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce a refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial-boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem sets isomorphisms between appropriate spaces pertaining to this scale.

Key words: Parabolic problem, interpolation with a function parameter, anisotropic Sobolev space, space of generalized smoothness, refined Sobolev scale, slowly varying function, isomorphism property.

### Article Information

 Title Parabolic problems and interpolation with a function parameter Source Methods Funct. Anal. Topology, Vol. 19 (2013), no. 2, 146-160 MathSciNet MR3098493 zbMATH 1289.35147 Milestones Received 02/02/2013 Copyright The Author(s) 2013 (CC BY-SA)

### Authors Information

Valerii Los
Department of Higher and Applied Mathematics, Chernigiv State Technological University, 95 Shevchenka, Chernigiv, 14027, Ukraine

Aleksandr A. Murach
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Valerii Los and Aleksandr A. Murach, Parabolic problems and interpolation with a function parameter, Methods Funct. Anal. Topology 19 (2013), no. 2, 146-160.

### BibTex

@article {MFAT689,
AUTHOR = {Los, Valerii and Murach, Aleksandr A.},
TITLE = {Parabolic problems and interpolation with a function parameter},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {2},
PAGES = {146-160},
ISSN = {1029-3531},
MRNUMBER = {MR3098493},
ZBLNUMBER = {1289.35147},
URL = {http://mfat.imath.kiev.ua/article/?id=689},
}