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

Abstract

AbstractThe Cauchy problem in $$\mathbb {R}^n$$ R n is considered for the Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ u t = Δ u - ∇ · ( u ∇ v ) , 0 = Δ v + u , ( ⋆ ) with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data $$u_0$$ u 0 with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$ u 0 is continuous and bounded, then ($$\star $$ ⋆ ) admits a local-in-time classical solution, whereas if $$u_0(x)\rightarrow +\infty $$ u 0 ( x ) → + ∞ as $$|x|\rightarrow \infty $$ | x | → ∞ , 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.