Existence and Stability for the Cauchy Problems of Nonlocal Keller–Segel Equations in Pseudomeasure Space

In this paper, we consider the Cauchy problem of the chemotaxis Keller–Segel equation with nonlocal operator $$$ {\left(-\Delta \right)}^{\frac{\beta }{2}}\kern0.3em \left(1<\beta \le 2\right) $$$ on the whole space  $$$ {\mathbb{R}}^d\kern0.3em \left(d\ge 1\right) $$$. In this paper, the global existence and local existence of mild solutions of nonlocal parabolic–parabolic Keller–Segel equations and nonlocal parabolic–elliptic Keller–Segel equations in pseudomeasure spaces are obtained. In addition, the stability of nonlocal Keller–Segel equations is established, that is, when the diffusion parameter $$$ \tau $$$ in the nonlocal parabolic–parabolic Keller–Segel equation tends to zero, the global mild solution of the nonlocal parabolic–parabolic Keller–Segel system converges to the global mild solution of the nonlocal parabolic–elliptic Keller–Segel system.