Well-posedness of a non-local abstract Cauchy problem with a singular integral

Abstract

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some bounded-ness and smoothness conditions on the coeffcient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with inflow boundary conditions.

Keywords

Partial integro-differential equation (PIDE) / singular integral / well-posedness / Wigner equation

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