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THE BOUNDARY VALUE PROBLEM FOR QUASILINEAR PARABOLIC EQUATIONS WITH A LEVY LAPLACIAN FOR FUNCTIONS OF INFINITE NUMBER OF VARIABLES Kovtun I.I. National University of Life and Environmental Sciences of Ukraine, Geroiv Oborony 15, Kiev 03041 Ukraine

ґ The Levy Laplacian of F (x) it the point x0 is defined (if it exists) by the formula [1] 1 LF (x0 ) = limn n n=1 (F (x0 )fk , fk )H , where function F (x) defined on the Hilbert space k H is twice strongly differentiable at a point x0 , F (x) is the Hessian of F (x), and {fk } 1 is an orthonormal basis in H . Let = be a domain in H , = {x H : 0 Q (x) < R2 }, is boundary and = {x H : Q (x) = R2 }. The function Q (x) is a twice strongly differentiable function 2 2 such that LQ (x) = , > 0 is positive constant. Consider the function T (x) = R -Q possesses the following properties 0 < T (x) R , LT (x) = -1 if x , and T (x) = 0 if x . Let in a certain functional class F exists a solution of the boundary value problem ( for the heat equations Vtt,x) = LV (t , x), V (t , x) = G(t, x) on , where G(t, x) is a given function defined on H . Consider the boundary value problem [2] U (t , x) = LU (t, x) + f0 (U (t, x)), t U (t, x) = G(t , x) on , (1) (2)
2

where U (t, x) is a function on [0, T ] в H , f0 () is a given function of one variable and exist both primitive () = f0d( ) and its inverse function -1 . Then solution of the boundary value problem (1), (2) is given by the equation U (t, x) = -1 (T (x) + (V (t, x))), where V (t , x) is the solution of the boundary value problem for the heat equations. References ґ 1. Levy P. Problemes concrets d'analyse fonctionnelle. Paris: Gauthier­Villars, 1951. ` 2. Feller M. N. , Kovtun I. I. Quasilinear parabolic equations with a Levy Laplacian for ` functions of infinite number of variables, Methods of functional analysis and topology. Volume 14. Number 2, 2008, PP. 117-123.