Single-point blow-up in the Cauchy problem for the higher-dimensional Keller–Segel system
Michael Winkler
Abstract
Abstract The Cauchy problem in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> for the Keller–Segel system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mfenced close="" open="{"> <mml:mrow> <mml:mtable class="cases"> <mml:mtr> <mml:mtd columnalign="left"> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mo>∇</mml:mo> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∇</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd columnalign="left"> <mml:msub> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mo>−</mml:mo> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:math> is considered for n ⩾ 3. Using a basic theory of local existence and maximal extensibility of classical and spatially integrable solutions as a starting point, the study provides a result on the occurrence of finite-time blow-up within considerably large sets of radially symmetric initial data, and moreover verifies that any such explosion exclusively occurs at the spatial origin. The detection of blow-up is accomplished by analyzing a relative of the well-known Keller–Segel energy inequality, involving a modification of the corresponding energy functional which, unlike the latter, can be seen to be favourably controlled from below by the corresponding dissipation rate through a certain functional inequality along trajectories.