Fractional nonlinear stochastic heat equation with variable thermal conductivity

Miloš Japundžić; Danijela Rajter-Ćirić

doi:10.1515/fca-2020-0087

Article

Fractional nonlinear stochastic heat equation with variable thermal conductivity

Miloš Japundžić and Danijela Rajter-Ćirić

Published/Copyright: December 31, 2020

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From the journal Fractional Calculus and Applied Analysis Volume 23 Issue 6

Abstract

We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).

MSC 2010: 26A33; 35R11; 46F30; 60G20

Key Words and Phrases: fractional Riesz derivative; stochastic heat equation; Colombeau generalized stochastic process; Colombeau generalized uniformly continuous semigroup of operators; regularized derivative

Editorial Note: This paper has been presented at the online international conference “FCTFA 2020: Topics in Fractional Calculus and Time-frequency Analysis”, University of Novi Sad, Serbia, 16–17 June 2020.

Acknowledgements

The paper is under the bilateral project between the Bulgarian Academy of Sciences and the Serbian Academy of Sciences and Arts through the project Operators, Differential Equations and Special Functions of Fractional Calculus - Numerics and Applications.

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Received: 2020-06-16

Published Online: 2020-12-31

Published in Print: 2020-12-16

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https://doi.org/10.1515/fca-2020-0087

Keywords for this article

fractional Riesz derivative; stochastic heat equation; Colombeau generalized stochastic process; Colombeau generalized uniformly continuous semigroup of operators; regularized derivative