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Abstracts

XVII conference

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

The Lévy Laplacian of $\;F(x)$ it the point $\;{x_0}\;$ is defined (if it exists) by the formula [1] $\Delta_LF(x_0)=\lim_{n \to \infty}\frac1n\sum_{k=1}^{n}(F''(x_0)f_k,f_k)_H, $ where function $F(x)$ defined on the Hilbert space $\;H\;$ is twice strongly differentiable at a point $\;{x_0},\;$ $\;F''(x)$ is the Hessian of $\;F(x),\; $ and $\{f_k\}_1^{\infty} $ is an orthonormal basis in $\;H.$

Let $\; \overline {\Omega}=\Omega \cup \Gamma$ be a domain in $\;H,$ $\Omega=\{x\in H: 0\le Q(x)<R^2\},$ $\;\Gamma\;$ is boundary and $\;\Gamma=\{x\in H:Q(x)=R^2\}.$ The function $\;Q(x)\; $ is a twice strongly differentiable function such that $\Delta _LQ(x)=\gamma,  \gamma >0\;$ is positive constant. Consider the function $T(x)=\frac{R^2-Q^2}{\gamma}$ possesses the following properties $\;0<T(x)\le \frac {R^2}{\gamma},\quad \Delta_LT(x)=-1\;$ if $\;x \in \Omega,\;$ and $\;T(x)=0\;$ if $\;x \in \Gamma.$ Let in a certain functional class exists a solution of the boundary value problem for the heat equations $\frac{\partial V(t,x)}{\partial t}=\Delta_LV(t,x), $   $ V(t,x)=G(t,x)$   on $ \Gamma, $ where $\;G(t,x)\;$ is a given function defined on $\;H.$

Consider the boundary value problem [2] $$\frac{\partial U(t,x)}{\partial t}=\Delta_LU(t,x))+f_0(U(t,x)),      (1)$$ $$ U(t,x)=G(t,x) \quad on \; \Gamma,           (2)$$ where $\;U(t,x)$ is a function on $\;[0, T]\times H,\;$ $\;f_0(\xi)$ is a given function of one variable and exist both primitive $\;\varphi(\xi)=\int \frac {d\xi}{f_0(\xi)}\;$ and its inverse function $\;\varphi^{-1}.$

Then solution of the boundary value problem (1), (2) is given by the equation $$U(t,x)=\varphi^{-1}(T(x)+\varphi(V(t,x))), $$ where $\;V(t,x)\;$ is the solution of the boundary value problem for the heat equations.

References

1. Lévy P. Problèmes concrets d'analyse fonctionnelle. Paris: Gauthier-Villars, 1951

2. Feller M. N. , Kovtun I. I. Quasilinear parabolic equations with a Lévy Laplacian for functions of infinite number of variables. Methods of functional analysis and topology. Volume 14. Number 2, 2008, PP. 117-123

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