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

DOI:
https://doi.org/10.1090/spmj/1596

Published electronically:
February 4, 2020

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

Email:
arinaark@gmail.com, arina@AA1101.spb.edu

DOI:
https://doi.org/10.1090/spmj/1596

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