Geodesic
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
Algebra i analiz, Tome 31 (2019) no. 2, pp. 118-151.

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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.
Keywords: parabolic systems, strong nonlinearity, global solvability. 
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A. A. Arkhipova. 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. Algebra i analiz, Tome 31 (2019) no. 2, pp. 118-151. http://geodesic.mathdoc.fr/item/AA_2019_31_2_a4/

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