Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity <jats:title>Abstract</jats:title><jats:p>The Cauchy problem in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {R}^n$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> is considered for the Keller–Segel system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\begin{aligned} \left\{ \begin{array}{l}u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>·</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mi>∇</mml:mi> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> <mml:mspace /> <mml:mspace /> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>⋆</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> with non-integrable behavior at spatial infinity. It is shown that if <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is continuous and bounded, then (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\star $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋆</mml:mo> </mml:math></jats:alternatives></jats:inline-formula>) admits a local-in-time classical solution, whereas if <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0(x)\rightarrow +\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> as <jats:inline-formula><jats:alternatives><jats:tex-math>$$|x|\rightarrow \infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, 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 (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\star $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋆</mml:mo> </mml:math></jats:alternatives></jats:inline-formula>) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.</jats:p> 9 2 919-959 919-959 Springer Science and Business Media LLC