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BOUNDARY VALUE PROBLEM FOR QUASILINEAR PARABOLIC EQUATIONS WITH A LEVY LAPLACIAN Kovtun I. I. P.O.B. 68, Kiev 04212 Ukraine

Let H be a real infinite dimensional Hilbert space. Let a scalar function F depend on ґ H is twice strongly differentiable at a point x0 . The Levy Laplacian of F at the point x0 is defined the formula [1] 1 LF (x0 ) = lim n n
n

(F (x0 )fk , fk )H ,
k=1

where F (x) is the Hessian of F (x), and {fk } is an orthonormal basis in H . 1 Let be a bounded domain in the Hilbert space H (that is a bounded open set in H ), and = be a domain in H with boundary : = {x H : 0 Q (x) R2 }, = {x H : Q (x) = R2 },

where Q (x) is a twice strongly differentiable function such that LQ (x) = , 0 is a positive constant. Consider the Cauchy problem U (t , x) = LU (t, x) + f0 (U (t, x)), t U (0, x) = U0 (x), (1)

where U (t , x) is a function on [0, T] в H , f0 () is a given function of one variable, U0 (x) is a given function defined on H . Assume exists a primitive () = f0d( ) and the inverse function -1 . Assume exists a solution of the Cauchy problem for the heat equation V (t , x) = LV (t , x), t V (0, x) = U0 (x).

Then the solution U (t , x) of the Cauchy problem (1) is U (t , x) = -1 (t + (V (t, x))). References. ґ 1. Levy P. Sur la generalisation de lequation de Laplace dans domaine fonctionnelle. ґ C.R.Acad. Sc. 168, 1919. P. 752-755.