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ON BOUNDARY PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS WITH LEVY LAPLACIAN Kovtun I.I. National University of Life and Environmental Sciences of Ukraine, Learning-scientific institute of energetic and automatic, Department higher and applied mathematic, 15, Geroiv Oborony, Kyiv, 03041, Ukraine e-mail: ira@otblesk.com Let H be a separable real Hilbert space. Let x H : x



2 H

R

2



be the ball

in H . Let U (t, x ) be the function in [0, ) , and LU (t, x ) be the Levy Laplacian [1], [2]. Consider the boundary value problem for nonlinear equations with Levy Laplacian U (t, x ) (1) f ( LU (t, x )) in , U (t, x ) G(t, x ) , t where f ( ) is a given continuous twice differentiable function. The equation f ( ) z can be solved with respect to : ( z ) . G(t, x ) is a given function, The solution of problem (1) exist, when exist solution V (t, x ) of boundary problem for V ( , x ) the heat equation LV ( , x ) in , V (t, x ) G(t, x ) . t V ( , x ) [t X ] T ( x ) 0 can be solved Theorem, Let the equation f X T ( x ) 1 2 with respect to X (t, x) , and (t, x ) t , T ( x ) R 2 x H . Then the solution 2 boundary problem (1) is





U (t, x) f ( (t, x))[t (t, x)] ( (t, x))T ( x) V (t, x) T ( x), x ,
V ( , x ) where ( (t , x )) .

( t , x ) T ( x )

References. 1. Levy P. Problemes concrets d'analyse fonctionnelle. - Paris: G.-V. 1951. 510 p. 2. Feller M.N. The Levy Laplacian. -Cambridge etc.: Cambridge Univ. Press. 2005. 153 p.