On an initial-boundary value problem for a parabolic pseudodifferential equation
Abstract
The article is aimed at investigation of a third initial-boundary value problem for a parabolic pseudo-differential equation that is related to an isotropic $\alpha$-stable stochastic process in multidimensional Euclidean space $\mathbb{R}^d$. The equation is linear with constant coefficients {with respect to} the partial derivative in time of unknown function and its fractional Laplacian of the order $\alpha \in (1,2)$.
The boundary condition is formed on a bounded closed surface that is sufficiently smooth. It equates a liner combination of inside and outside limits of the pseudo-derivative of order $\beta\in (0, \alpha-1)$ in the normal direction to the surface, and the value of the unknown function itself. We have established some estimates of the fundamental solutions of the given problem.
References
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