Opuscula Math. 34, no. 2 (2014), 425-441
http://dx.doi.org/10.7494/OpMath.2014.34.2.425

Opuscula Mathematica

# On the Tonelli method for the degenerate parabolic Cauchy problem with functional argument

Krzysztof A. Topolski

Abstract. The degenerate parabolic Cauchy problem is considered. A functional argument in the equation is of the Hale type. As a limit of piecewise classical solutions we obtain a viscosity solution of the main problem. Presented method is an adaptation of Tonelli's constructive method to the partial differential-functional equation. It is also shown that this approach can be improved by the vanishing viscosity method and regularisation process.

Keywords: viscosity solutions, parabolic equation, differential-functional equation.

Mathematics Subject Classification: 35A01, 35K15, 35K60.

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