Local and global solutions for a subdiffusive parabolic-parabolic Keller-Segel system (Q6612984)

This is a study of the parabolic system of Keller-Segel type with time fractional derivatives. The appearance of time nonlocal terms in the fractional derivatives of distinct order in these two equations leads to a subdiffusive behavior or a kind of memory in that model of chemotaxis. Using the framework of mild solutions and the Laplace transform, a couple of well-posedness results, as well as asymptotic behavior of solutions and a criterion for continuation of solutions (i.e. non-blow-up condition) are shown. These are extensions of the studies of the model with the time derivatives of the same order begun in [\textit{J. Azevedo} et al., Math. Nachr. 292, No. 3, 462--480 (2019; Zbl 1419.35200); \textit{C. Cuevas} et al., Math. Methods Appl. Sci. 43, No. 2, 769--798 (2020; Zbl 1445.35301)].