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Архив публикацийТезисыXIX-ая конференцияО краевой задаче для нелинейного параболического уравнения с лапласианом ЛевиУкраина, 04212, г. Киев, ул. Малиновского, 11, кв. 399 1 стр. (принято к публикации)Let $H$ be a separable real Hilbert space. Let $\; \overline {\Omega}=\Omega \cup \Gamma=\{x\in H: \|x\|_H^2\le R^2\}$ be the ball in $\;H.$ Let $U(t,x)$ be a function on $[0,\infty)\times \Omega.$ $\Delta_LU(t,x)$ is the L\`{e}vy Laplacian [1], [2].
Consider the boundary value problem for nonlinear parabolic equations with L\`{e}vy Laplacian $$\frac{\partial U(t,x)}{\partial t}=f(\Delta_LU(t,x)) \quad in \quad \Omega \quad U(t,x)=G(t,x) \quad on \quad \Gamma,\eqno(1)$$ where $\;f(\xi)$ is a given continuous twice differentiable function. The equation $\;f(\xi)=z\;$ can be solved with respect to $\xi: \;$ $ \xi=f(z).\;$ The function $\;G(t,x)\;$ is a given function.
The solution of problem (1) exist, when exist solution $V(\tau,x)\;$ of boundary problem for the heat equations $$\frac{\partial V(\tau,x)}{\partial \tau}=\Delta_LV(\tau,x)), \quad in \quad \Omega,\quad V(\tau,x)=G(\tau,x) \quad on \quad \Gamma, $$ where $\;G(\tau,x)\;$ is a given function defined on $\;H.$
{\bf Theorem}. Let the equation $$f'\Bigl(\varphi\Bigl(\frac {\partial V(\tau,x)}{\partial \tau}\Bigl|_{\tau=X+T(x)}\Bigl)\Bigl)[t-X]-T(x)=0$$can be solved with respect to $X=\chi(t,x),$ and $\;\chi(t,x)\Bigl|_{\Gamma}=t, \; \; T(x)=\frac{R^2-\|x\|_H^2}2.$ Then the solution value boundary problem (1) is$$U(t,x)=f\Bigl(\psi(t,x)\Bigl)[t-\chi(t,x)]-\psi\Bigl(\chi(t,x)\Bigl)T(x)+V((\chi(t,x))+ T(x),x),$$ where $\;\psi\chi(t,x))=\varphi\Bigl(\frac{\partial V(\tau,x)}{\partial \tau}\Bigl|_{\tau=\chi(t,x)+T(x)}\Bigl).$
\begin{center} {\bf References } \end{center} \begin{enumerate}
% 1 \item {\it L\'{e}vy~P. $\;$} Probl\`{e}mes concrets d'analyse fonctionnelle. Paris: Gauthier--Villars, 1951.
% 2 \item {\it Feller~M.~N.$\;$} The L\`{e}vy Laplacian. -Cambridge etc.: Cambridge Univ. Press. 2005. 153p.
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