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09 February 2025
 
  » arxiv » 1508.0117

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Well-posedness and Gevrey Analyticity of the Generalized Keller-Segel System in Critical Besov Spaces
Jihong Zhao ;
Date 1 Aug 2015
AbstractIn this paper, we study the Cauchy problem for the generalized Keller-Segel system with the cell diffusion being ruled by fractional diffusion: egin{equation*} egin{cases}
partial_{t}u+Lambda^{alpha}u- ablacdot(u abla psi)=0quad &mbox{in}
mathbb{R}^n imes(0,infty),
-Delta psi=uquad &mbox{in}
mathbb{R}^n imes(0,infty),
u(x,0)=u_0(x), &mbox{in} mathbb{R}^n. end{cases} end{equation*} In the case that $1<alphaleq 2$, we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces $dot{B}^{-alpha+frac{n}{p}}_{p,q}(mathbb{R}^{n})$ with $1leq p<infty$, $1leq qleq infty$, and analyticity of solutions for initial data $u_{0}in dot{B}^{-alpha+frac{n}{p}}_{p,q}(mathbb{R}^{n})$ with $1< p<infty$, $1leq qleq infty$. Moreover, the global existence and analyticity of solutions with small initial data in critical Besov spaces $dot{B}^{-alpha}_{infty,1}(mathbb{R}^{n})$ is also established. In the limit case that $alpha=1$, we prove global well-posedness for small initial data in critical Besov spaces $dot{B}^{-1+frac{n}{p}}_{p,1}(mathbb{R}^{n})$ with $1leq p<infty$ and $dot{B}^{-1}_{infty,1}(mathbb{R}^{n})$, and show analyticity of solutions for small initial data in $dot{B}^{-1+frac{n}{p}}_{p,1}(mathbb{R}^{n})$ with $1<p<infty$ and $dot{B}^{-1}_{infty,1}(mathbb{R}^{n})$, respectively.
Source arXiv, 1508.0117
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