Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
Michael Winkler
Abstract
Abstract The Cauchy problem in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{n} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> n\ge 2 , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="right"> <m:mfenced open="{" close=""> <m:mrow> <m:mspace depth="1.25em"/> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mo>⋅</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mi>S</m:mi> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mn>0</m:mn> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>v</m:mi> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> <m:mspace width="2.0em"/> <m:mspace width="2.0em"/> <m:mspace width="2.0em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>⋆</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:math> \begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> S\in {{\mathbb{R}}}^{n\times n} . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">BUC</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some <jats: