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

This work is concerned with the fractional-in-time parabolic-parabolic Keller-Segel system in a bounded domain Omega subset of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}<^>{d}$$\end{document} (d >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document}), for distinct fractional diffusions of the cells and chemoattractant. We prove results on existence, uniqueness, continuous dependence on the initial data and its robustness, continuation, and a blow-up alternative of solutions in Lebesgue spaces. Then, we use those results to show the existence of global solutions to the problem, when the chemoattractant diffusion is not slower than the cell diffusion.