# American Institute of Mathematical Sciences

January  2017, 16(1): 69-98. doi: 10.3934/cpaa.2017003

## 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

Received  November 2015 Revised  July 2016 Published  November 2016

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.

Citation: Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003
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