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

In this paper, the local well‐posedness (which means that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ with 1≤p≤+∞,1≤r<+∞$$ 1\le p\le &#x0002B;\infty, 1\le r&lt;&#x0002B;\infty $$ and s>1+dp$$ s&gt;1&#x0002B;\frac{d}{p} $$ was established, and the solution of PEKS (Keller‐Segel system with β>0$$ \beta &gt;0 $$) converges to the solution of HEKS (Keller‐Segel system with β=0$$ \beta &#x0003D;0 $$) as the diffusion coefficient β→0$$ \beta \to 0 $$ 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 Bp,11+dp(ℝd)$$ {B}_{p,1}&#x0005E;{1&#x0002B;\frac{d}{p}}\left({\mathbb{R}}&#x0005E;d\right) $$ but ill‐posed (not continuous) in Besov spaces B2,∞s(ℝd)$$ {B}_{2,\infty}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$. Moreover, we show a further continuity result that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is Hölder continuous in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ was also obtained. [ABSTRACT FROM AUTHOR]