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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Weak global solvability of the two-phase problem for a class of parabolic systems with strong nonlinearity in the gradient. The case of two spatial variables

Author: A. A. Arkhipova
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 273-296
MSC (2010): Primary 35K59
Published electronically: February 4, 2020
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Abstract: A class of quasilinear parabolic systems with nondiagonal principal matrix and strongly nonlinear additional terms is considered. The elliptic operator of the system has a variational structure and is generated by a quadratic functional with a nondiagonal matrix. A plane domain of the spatial variables is divided by a smooth curve in two subdomains and the principal matrix of the system has a ``jump'' when crossing this curve. The two-phase conditions are given on this curve and the Cauchy-Dirichlet conditions hold at the parabolic boundary of the main parabolic cylinder. The existence of a weak Hölder continuous global solution of the two-phase problem is proved. The problem can be regarded as a construction of the heat flow from a given vector-function to an extremal of the functional.

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Additional Information

A. A. Arkhipova
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Keywords: Parabolic systems, strong nonlinearity, global solvability
Received by editor(s): November 30, 2018
Published electronically: February 4, 2020
Additional Notes: The author’s research has been financially supported by the Russian Foundation for Basic Research (RFBR), grant no. 18-01-00472
Dedicated: Dedicated to Vladimir G. Maz’ya
Article copyright: © Copyright 2020 American Mathematical Society