# The Cauchy problem for the heat equation on the plane with a random right part from the Orlicz space

## DOI:

https://doi.org/10.17721/1812-5409.2020/3.11## Abstract

The heat equation with random conditions is a classical problem of mathematical physics. Recently, a number of works appeared, which in many ways investigated this equation according to the type of random initial conditions. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on the plane with a random right part. We consider the right part as a random function of the Orlicz space. The conditions of existence with probability one classical solution of the problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

* Key words*: stochastic processes of the Orlicz space, heat equation, estimation for the distribution of the supremum solution.

* Pages of the article in the issue*: 103 - 109

** Language of the article**: Ukrainian

## References

RATANOV N. E., SHUHOV A. G., SUHOV Yu. M. (1991) “Stabilization of the statistical solution of the parabolic equation” , Acta Appl. Math., 22, pp. 103–115.

WOYCZYNSKI W. A.(1998) Burgers-KPZ Turbulence “Lecture Notes in Math”. Springer Verlag, Berlin, Heidelberg., Vol. 1700.

SURGAILIAS D.,WOYCZYNSKI W. A. (2003) “Limit theorems for the Burgers equation initialized by data withlong-range dependence Theory and Applications of Longrange Dependence, Birkhausser, Boston.

KOZACHENKO Yu. V., LEONENKO G. M. (2006) “Extremal behavior of the heat random field”, Extremes, 8, pp. 191–205.

BEGHIN L., KOZACHENKO Yu. V., ORSINGHER E., SAKHNO L. (2007) “On the solution of linear odd-order heat-type equations with random initial”, Journal of Statistical Physics., Vol. 127, No. 4. P. 721–739.

KOZACHENKO Yu. V., VERESH K. J. (2009) “The heat equation with random initial conditions from Orlicz space”, Teor. Imovirnost. Matem. Statist., 8, pp. 63–75.

KOZACHENKO Yu. V., VERESH K. J. (2010) “Boundary-value problem for nonhomogeneousparabolic equation with Orlicz right side” Random Operators and Stochastic Equations 18., pp. 97–119.

KOZACHENKO Yu. V., SLYVKA-TYLYSHCHAK A. I. (2014) “The Cauchy problem for the heat equation with a random right part from the space Sub_φ(Ω)”, Applied Mathematics, 5, pp. 2318–2333.

KOZACHENKO Yu. V., SLYVKA-TYLYSHCHAK A. I. (2014) “The Cauchy problem for the heat equation with a random right side”, Random Oper. and Stoch. Equ., 22(1), pp. 53–64.

SLYVKA-TYLYSHCHAK A. I. (2014) “The heat equation on line with random right part from Orlicz space”, Carpatian Mathematical Publications, 6 no. 1., pp. 134-148.

BULDYGIN V. V., KOZACHENKO Yu. V. (2000) “Metric Characterization of Random Variables and Random processes”, American Mathematical Society, Providence, Rhode, 285 p.

MARKOVICH B. M. (2010) “Equations of Mathematical Physics”, Lviv: Lviv Polytechnic Publishing House, 2 384 p.

KOZACHENKO Yu. V., SLYVKA G. I. (2003) “Justification of the Fourier method for hyperbolic equations with random initial conditions”, Theory Probab. and Mathem. Statist., 69, pp. 67–83.

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (3), 103–109. https://doi.org/10.17721/1812-5409.2020/3.11

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