On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
Anh Tuan Nguyen, Nguyen Huy Tuan, Chao Yang
Abstract
Abstract In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{d} , a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.