Властивостi узагальненого розв’язку задачi Кошi для рiвняння теплопровiдностi з випадковою правою частиною

Abstract

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet. In the works by E. Beisenbaev, Yu.V. Kozachenko and V.V. Buldygin a new approach studying the solutions of partial differential equations with random initial conditions was proposed. The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem. The mentioned approach was used in the worksby E. Barrasa de La Krus, Endzhyrgly, Ya.A. Kovalchuk. In the paper by V.V. Buldygin and Yu.V. Kozachenko the application of the Fourier method for the homogeneous hyperbolic equation with Gaussian initial conditions is justified and existence conditions in terms of correlation functions are studied. Homogeneous hyperbolic equation with random initial conditions from the space Sub ϕ (Ω) are considered in works by B. V. Dovgay, G.I.Slyvka-Tylyshchak. The model of a solution of a hyperbolic type equation with random initial conditions was investigated in the papers by G.I. Slyvka-Tylyshchak. There is studied a boundary value problem of mathematical physics for the inhomogeneous hyperbolic equation with ϕ-subgaussian in right part in works by B. V. Dovgay. The parabolic typeequations of Mathematical Physics with random factors of Orlicz spaces have been studied in the papers by Yu.V. Kozachenko and K.J. Veresh. Properties of the classical solution of the heat equation on a line with a random right part are considered in works by Yu.V. Kozachenko and G.I. Slyvka-Tylyshchak.We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space L p (Ω). The conditions of existence with probability one generalized solution of the problem are investigated.