Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type

<jats:title>Abstract</jats:title> <jats:p>The Cauchy problem in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_001.png" /> <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> <jats:tex-math>{{\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_002.png" /> <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> <jats:tex-math>n\ge 2</jats:tex-math> </jats:alternatives> </jats:inline-formula>, for <jats:disp-formula id="j_math-2022-0578_eq_001"> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_003.png" /> <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> <jats:tex-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}</jats:tex-math> </jats:alternatives> </jats:disp-formula> is considered for general matrices <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_004.png" /> <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> <jats:tex-math>S\in {{\mathbb{R}}}^{n\times n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_005.png" /> <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> <jats:tex-math>{\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> with some <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_006.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>p\in \left[1,n)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, there exist <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_007.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>{T}_{\max }\in \left(0,\infty ]</jats:tex-math> </jats:alternatives> </jats:inline-formula> and a uniquely determined <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_008.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>;</m:mo> <m:mspace width="0.33em" /> <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:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>;</m:mo> <m:mspace width="0.33em" /> <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:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>∞</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:mo>×</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }))</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_009.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> <m:mo>≔</m:mo> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⋆</m:mo> <m:mi>u</m:mi> </m:math> <jats:tex-math>v:= \Gamma \star u</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_010.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> <jats:tex-math>\Gamma </jats:tex-math> </jats:alternatives> </jats:inline-formula> denoting the Newtonian kernel on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_011.png" /> <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> <jats:tex-math>{{\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_012.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\left(u,v)</jats:tex-math> </jats:alternatives> </jats:inline-formula> forms a classical solution of (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_013.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>⋆</m:mo> </m:math> <jats:tex-math>\star </jats:tex-math> </jats:alternatives> </jats:inline-formula>) in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_014.png" /> <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:mo>×</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which has the property that <jats:disp-formula id="j_math-2022-0578_eq_002"> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_015.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mspace width="0.1em" /> <m:mtext>if</m:mtext> <m:mspace width="0.1em" /> <m:mspace width="0.33em" /> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> <m:mo>&lt;</m:mo> <m:mi>∞</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em" /> <m:mstyle> <m:mspace width="0.1em" /> <m:mtext>then both</m:mtext> <m:mspace width="0.1em" /> </m:mstyle> <m:mspace width="0.33em" /> <m:munder> <m:mrow> <m:mi>limsup</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>↗</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> </m:munder> <m:msub> <m:mrow> <m:mo>‖</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>⋅</m:mo> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</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:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>∞</m:mi> <m:mspace width="1.0em" /> <m:mspace width="0.1em" /> <m:mtext>and</m:mtext> <m:mspace width="0.1em" /> <m:mspace width="1.0em" /> <m:munder> <m:mrow> <m:mi>limsup</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>↗</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>max</m:mi> </m:mrow> </m:msub> </m:mrow> </m:munder> <m:msub> <m:mrow> <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:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</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:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>∞</m:mi> <m:mo>.</m:mo> </m:math> <jats:tex-math>\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty .</jats:tex-math> </jats:alternatives> </jats:disp-formula> An exemplary application of this provides a result on global classical solvability in cases when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_016.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∣</m:mo> <m:mi>S</m:mi> <m:mo>+</m:mo> <m:mn mathvariant="bold">1</m:mn> <m:mo>∣</m:mo> </m:math> <jats:tex-math>| S+{\bf{1}}| </jats:tex-math> </jats:alternatives> </jats:inline-formula> is sufficiently small, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0578_eq_017.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn mathvariant="bold">1</m:mn> <m:mo>=</m:mo> <m:mi mathvariant="normal">diag</m:mi> <m:mspace width="0.33em" /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1)</jats:tex-math> </jats:alternatives> </jats:inline-formula>.</jats:p>