A degenerate migration-consumption model in domains of arbitrary dimension

<jats:title>Abstract</jats:title> <jats:p>In a smoothly bounded convex domain <jats:inline-formula id="j_ans-2023-0131_ineq_001"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Ω</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:mrow> </m:msup> </m:math> <jats:tex-math> ${\Omega}\subset {\mathbb{R}}^{n}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_ineq_001.png"/> </jats:alternatives> </jats:inline-formula> with <jats:italic>n</jats:italic> ≥ 1, a no-flux initial-boundary value problem for<jats:disp-formula id="j_ans-2023-0131_eq_999"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <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:mfenced close=")" open="("> <m:mrow> <m:mi>u</m:mi> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>v</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>v</m:mi> <m:mo>−</m:mo> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math> $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ </jats:tex-math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_eq_999.png"/> </jats:alternatives> </jats:disp-formula>is considered under the assumption that near the origin, the function <jats:italic>ϕ</jats:italic> suitably generalizes the prototype given by<jats:disp-formula id="j_ans-2023-0131_eq_998"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="2em"/> <m:mi>ξ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> <jats:tex-math> $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ </jats:tex-math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_eq_998.png"/> </jats:alternatives> </jats:disp-formula>By means of separate approaches, it is shown that in both cases <jats:italic>α</jats:italic> ∈ (0, 1) and <jats:italic>α</jats:italic> ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy<jats:disp-formula id="j_ans-2023-0131_eq_997"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>C</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≔</m:mo> <m:munder> <m:mrow> <m:mtext>ess sup</m:mtext> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:munder> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>⋅</m:mo> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>ln</m:mi> <m:mo>⁡</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>⋅</m:mo> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>&lt;</m:mo> <m:mi>∞</m:mi> <m:mspace width="2em"/> <m:mtext>for all </m:mtext> <m:mi>T</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math> $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){&lt; }\infty \qquad \text{for\,all\,}T{ &gt;}0,$$ </jats:tex-math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_eq_997.png"/> </jats:alternatives> </jats:disp-formula>with sup<jats:sub> <jats:italic>T</jats:italic>&gt;0</jats:sub> <jats:italic>C</jats:italic>(<jats:italic>T</jats:italic>) &lt; ∞ if <jats:italic>α</jats:italic> ∈ [1, 2].</jats:p>