On the Cauchy problem for Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces

Abstract

In this paper, the local well‐posedness (which means that the initial‐to‐solution map is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces with and was established, and the solution of PEKS (Keller‐Segel system with ) converges to the solution of HEKS (Keller‐Segel system with ) as the diffusion coefficient in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces but ill‐posed (not continuous) in Besov spaces . Moreover, we show a further continuity result that the initial‐to‐solution map is Hölder continuous in Besov spaces equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces was also obtained.