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 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 in ow boundary conditions.
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The authors are grateful to Prof. Ruo Li and Dr. Zhangpeng Sun for their helpful discussion. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671038, 91630130, 91434201, 11421101).
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Jiang, H., Lu, T. & Zhu, X. Well-posedness of a non-local abstract Cauchy problem with a singular integral. Front. Math. China 14, 77–93 (2019). https://doi.org/10.1007/s11464-019-0750-3
- Partial integro-differential equation (PIDE)
- singular integral
- Wigner equation