Well-posedness and Gevrey analyticity of the generalized Keller-Segel system in critical Besov spaces

In this paper, we study the Cauchy problem for the generalized Keller-Segel system with the cell diffusion being ruled by fractional diffusion: {partial derivative(t)u + Lambda(alpha)u + del center dot (u del psi) = 0 in R-n x (o, infinity), -Delta psi = u in R-n x (0, infinity) u(x, 0) = u(0) (x) in R-n. In the case 1 < alpha <= 2, we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces (B)over dot(p,q) (-alpha+n/p) (R-n) with 1 <= p < infinity, 1 <= q <= infinity , and analyticity of solutions for initial data u(0) is an element of(B)over dot(p,q) (-alpha+n/p) (R-n) with 1 < p < infinity,1 <= q <= infinity. Moreover the global existence and analyticity of solutions with small initial data in critical Besov spaces (B)over dot(infinity,1)(-alpha) (R-n) is also established. In the limit case alpha = 1, we prove global well-posedness for small initial data in critical Besov spaces (B)over dot(p,1) (-1+n/p) (R-n) with 1 <= p < infinity and (B)over dot(infinity,1)(-1) (R-n) and show analyticity of solutions for small initial data in (B)over dot(p,1) (-1+n/p) (R-n) with 1 < p < 8 and (B)over dot(infinity,1)(-1) (R-n), respectively.