Electron. J. Diff. Equ., Vol. 2015 (2015), No. 113, pp. 1-12.

Noncontinuous solutions to degenerate parabolic inequalities

Krzysztof A. Topolski

We consider the initial value problem for degenerate parabolic equations. We prove theorems on differential inequalities and comparison theorems in unbounded domain. As a solution of differential inequality we consider upper absolutely (lower absolutely) continuous in t function (we admit discontinuity in time variable). In the last section we compare our notion of subsolutions to the notion of viscosity subsolutions smooth in space variable. By giving a counterexample we show that upper absolutcontinuity plays crucial role in the equivalence of the two notions.

Submitted October 2, 2014. Published April 28, 2015.
Math Subject Classifications: 35D30, 35K51, 35R45.
Key Words: Parabolic equations; Cauchy problem; generalized solution.

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Krzysztof A. Topolski
Institute of Mathematics
University of Gdansk
Wit Stwosz 57, 80-952 Gdansk, Poland
email: matkt@mat.ug.edu.pl

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