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

Topolski K. A.

Opuscula Mathematica

2014

Vol. 34, no. 2

425--441

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.

On the unbounded solutions for parabolic differential-functional Cauchy problem

Topolski K.

Demonstratio Mathematica

2007

Vol. 40, nr 1

151-160

We consider the initial value problem for second order differential–func- tional equation. Functional dependence on an unknown function is of the Hale type. We prove the existence theorem for unbounded classical solution. Our formulation admits a large group of nonlocal problems. We put particular stress on “retarded and deviated” argument as it seems to be the most difficult.

On the classical solutions for parabolic differential - functional Cauchy problem

Topolski K. A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr 2

217-226

We consider the initial value problem for second order differential - functional parabolic equation. Functional dependence is of the Hale type. On the basis of differential inequalities and fixed point method we prove the existence theorem for classical solution. Our formulation covers a large group of nonlocal problems such as , integro-differential equations, and "retarded and deviated" argument. We put particular stress on the last one, as it requires more general treatment.