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arXiv · 1801.09843

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

H. Jiang·T. Lu·X. Zhu
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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 boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with 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.

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H. Jiang, T. Lu, X. Zhu. 2018-01-30. Well-posedness of a non-local abstract Cauchy problem with a singular integral. https://arxiv.org/abs/1801.09843

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