Existence and regularity of the fractional Keller–Segel equation via the extended theory of semigroups

This paper intends to look into the existence and regularity of the solution for the fractional Keller–Segel equation by virtue of the extended theory of semigroups, that is, the theory of resolvent families, which is one of the powerful tools for solving partial differential equations. First of all, we establish the theory of resolvent families with two parameters related to the fractional Keller–Segel equation, including the generation theorem and the estimations of resolvent families. Then, we research the existence and Hölder regularity results of the classical solution in fundamental spaces $L^2\left(T^N\right)$ and $L^p\left(R^N\right)$ by applying the theory of resolvent families with two parameters, respectively. Moreover, we attain some new $L^q$-$L^r$ estimations of the resolvent families by utilizing the Gagliardo–Nirenberg inequality, and investigate the local existence and integrability of the mild solution in fundamental spaces $L^q\left(R^N\right)$ and $L^r\left(R^N\right)$. In the end, we explore the global existence of the mild solution in fundamental spaces $L^{\frac{N}{\alpha +\beta -2}}\left(R^N\right)$ and $L^r\left(R^N\right)$. Compared to the results that derived from the perspective of partial differentiation, we possess some new meaningful results and findings. The results of this article can be served as supplements to those obtained using the methods of partial differentiation.