Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity

Abstract

The Cauchy problem in mathbb {R}^n is considered for the Keller–Segel system ut=Δu-∇·(u∇v),0=Δv+u,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\Delta u - \ abla \\cdot (u\ abla v), \\\\ 0 = \\Delta v + u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data u_0 with non-integrable behavior at spatial infinity. It is shown that if u_0 is continuous and bounded, then (star ) admits a local-in-time classical solution, whereas if u_0(x)rightarrow +infty as |x|rightarrow infty , then no such solution can be found. Furthermore, a collection of three sufficient criteria for either global existence or global nonexistence indicates that with respect to the occurrence of finite-time blow-up, spatial decay properties of an explicit singular steady state plays a critical role. In particular, this underlines that explosions in (star ) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.