An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

Valerii Los; Vladimir A. Mikhailets; Aleksandr A. Murach

doi:10.3934/cpaa.2017003

Article Contents

2017, Volume 16, Issue 1: 69-98. Doi: 10.3934/cpaa.2017003

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An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

1.
National Technical University of Ukraine "Kyiv Polytechnic Institute", Prospect Peremohy 37,03056, Kyiv-56, Ukraine
2.
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3,01004 Kyiv, Ukraine
3.
Chernihiv National Pedagogical University, Het'mana Polubotka str. 53,14013 Chernihiv, Ukraine

Author Bio: E-mail address: v_los@yahoo.com; E-mail address: mikhailets@imath.kiev.ua
E-mail address: murach@imath.kiev.ua
E-mail address: murach@imath.kiev.ua

Received: November 2015

Revised: July 2016

Published: January 2018

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Abstract

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

Keywords:

Parabolic initial-boundary value problem,
Hörmander space,
slowly varying function,
isomorphism property,
interpolation with a function parameter.

Mathematics Subject Classification: Primary: 35K35; Secondary: 46B70, 46E35.

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References

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