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Abstracts

XVI conference

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 $\;H\;$ is twice strongly differentiable at a point $\;x_0.\;$ The L\'evy Laplacian of $\;F\;$ at the point $\;x_0\;$ is defined 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 $\;F''(x)$ is the Hessian of $\;F(x),\;$ and $ \{f_k\}_1^{\infty} $ is an orthonormal basis in $\;H.$

Let $\;\Omega\;$ be a bounded domain in the Hilbert space $\;H\;$ (that is a bounded open set in $\;H$), and $\; \overline {\Omega}=\Omega \cup \Gamma\;$ be a domain in $\;H\;$ with boundary $\;\Gamma:$ $$\Omega=\{x\in H: 0\le Q(x)\langleR^2\}, \quad \Gamma=\{x\in H:Q(x)=R^2\},$$ where $\;Q(x)\; $ is a twice strongly differentiable function such that $\;\Delta _LQ(x)=\gamma,$ $\gamma\rangle0 \;$ is a positive constant.

Consider the Cauchy problem $$\frac{\partial U(t,x)}{\partial t}=\Delta_LU(t,x)+f_0(U(t,x)), \qquad U(0,x)=U_0(x), \eqno(1) $$ where $\;U(t,x)\;$ is a function on $\;[0,{\frak T}] \times H, \; f_0(\xi)\;$ is a given function of one variable, $\;U_0(x)\;$ is a given function defined on $H$.

Assume exists a primitive $\varphi(\xi)=\int \frac {d\xi}{f_0(\xi)}\;$ and the inverse function $\;\varphi^{-1}.$ Assume exists a solution of the Cauchy problem for the heat equation $$\frac{\partial V(t,x)}{\partial t}=\Delta_LV(t,x), \quad V(0,x)=U_0(x). $$ Then the solution $U(t,x)$ of the Cauchy problem (1) is $$U(t,x)=\varphi^{-1}(t+\varphi(V(t,x))).$$

References

1. L'evy P. Sur la generalisation de l'equation de Laplace dans domaine fonctionnelle. C.R.Acad. Sc. 1919. V. 168. P. 752-755.

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