On an initial-boundary value problem for a parabolic pseudodifferential equation
Анотація
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.
Посилання
M.M. Osypchuk, M.I. Portenko, Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudo-differential equation, Ukr. Math. J., 69 (2018), №10, 1631–1650. https://doi.org/10.1007/s11253-018-1459-2
M.M. Osypchuk, M.I. Portenko, On simple-layer potentials for one class of pseudodifferential equations, Ukr. Math. J., 67 (2016), №11, 1704–1720, https://doi.org/10.1007/s11253-016-1184-7
S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analytic methods in the theory of differential and pseudodifferential equations of parabolic type, Operator Theory Advances and Applications, V.152, Birkhauser Verlag, 2004. https://doi.org/10.1007/978-3-0348-7844-9
K. Mamalyha, M. Osypchuk, On single-layer potentials, pseudo-gradients and jump theorem for an isotropic stable stochastic process, J. Pseudo-Differ. Oper. Appl., 15 (4) (2024). https://doi.org/10.1007/s11868-023-00574-y
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