Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction

<jats:title>Abstract</jats:title><jats:p>A no-flux initial-boundary value problem for<jats:disp-formula id="nonace22eueqn1"><jats:tex-math><?CDATA \begin{align*} \begin{cases} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{cases} \qquad \qquad (\star) \end{align*}?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"><mml:mtable columnalign="right left right left right left right left right left right left" columnspacing="0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em" rowspacing="3pt"><mml:mtr><mml:mtd><mml:mfenced close="" open="{"><mml:mtable columnalign="left left" columnspacing="1em" rowspacing=".1em"><mml:mtr><mml:mtd><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mrow><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>v</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>v</mml:mi><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo stretchy="false">(</mml:mo><mml:mo>⋆</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" position="float" xlink:href="nonace22eueqn1.gif" xlink:type="simple" /></jats:disp-formula>is considered in smoothly bounded subdomains of<jats:inline-formula><jats:tex-math><?CDATA $\mathbb{R}^n$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn1.gif" xlink:type="simple" /></jats:inline-formula>with<jats:inline-formula><jats:tex-math><?CDATA $n\geqslant 1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn2.gif" xlink:type="simple" /></jats:inline-formula>and suitably regular initial data, where<jats:italic>φ</jats:italic>is assumed to reflect algebraic type cross-degeneracies by sharing essential features with<jats:inline-formula><jats:tex-math><?CDATA $0\leqslant \xi\mapsto \xi^\alpha$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mn>0</mml:mn><mml:mo>⩽</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">↦</mml:mo><mml:msup><mml:mi>ξ</mml:mi><mml:mi>α</mml:mi></mml:msup></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn3.gif" xlink:type="simple" /></jats:inline-formula>for some<jats:inline-formula><jats:tex-math><?CDATA $\alpha\geqslant 1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>α</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn4.gif" xlink:type="simple" /></jats:inline-formula>. Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profile<jats:inline-formula><jats:tex-math><?CDATA $u_\infty$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi>u</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msub></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn5.gif" xlink:type="simple" /></jats:inline-formula>and some null set<jats:inline-formula><jats:tex-math><?CDATA $N_\star\subset (0,\infty)$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi>N</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>⊂</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn6.gif" xlink:type="simple" /></jats:inline-formula>, a corresponding global generalized solution, known to exist according to recent literature, satisfies<jats:disp-formula id="nonace22eueqn2"><jats:tex-math><?CDATA \begin{align*} \rho(u(\cdot,t))\stackrel{\star}{\rightharpoonup} \rho(u_\infty) \quad \textrm{in } L^\infty(\Omega) \quad\;\; \textrm{ and } \quad\;\; v(\cdot,t)\to 0 \quad \textrm{in } L^p(\Omega)\; \textrm{for all } p\geqslant 1 \end{align*}?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"><mml:mtable columnalign="right left right left right left right left right left right left" columnspacing="0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em" rowspacing="3pt"><mml:mtr><mml:mtd><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mo stretchy="false">⇀</mml:mo></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mtext>in </mml:mtext></mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mtext> and </mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn><mml:mrow><mml:mtext>in </mml:mtext></mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mtext>for all </mml:mtext></mml:mrow><mml:mi>p</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" position="float" xlink:href="nonace22eueqn2.gif" xlink:type="simple" /></jats:disp-formula>as<jats:inline-formula><jats:tex-math><?CDATA $(0,\infty)\setminus N_\star \ni t\to \infty$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∋</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn7.gif" xlink:type="simple" /></jats:inline-formula>, where<jats:inline-formula><jats:tex-math><?CDATA $\rho(\xi): = \frac{\xi^2}{(\xi+1)^2}$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>:=</mml:mo><mml:mfrac><mml:msup><mml:mi>ξ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mfrac></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn8.gif" xlink:type="simple" /></jats:inline-formula>,<jats:inline-formula><jats:tex-math><?CDATA $\xi\geqslant 0$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>ξ</mml:mi><mml:mo>⩾</mml:mo><mml:mn>0</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn9.gif" xlink:type="simple" /></jats:inline-formula>. In the particular case when either<jats:inline-formula><jats:tex-math><?CDATA $n\leqslant 2$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩽</mml:mo><mml:mn>2</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn10.gif" xlink:type="simple" /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $\alpha\geqslant 1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>α</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn11.gif" xlink:type="simple" /></jats:inline-formula>is arbitrary, or<jats:inline-formula><jats:tex-math><?CDATA $n\geqslant 1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn12.gif" xlink:type="simple" /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $\alpha\in [1,2]$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn13.gif" xlink:type="simple" /></jats:inline-formula>, additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states<jats:inline-formula><jats:tex-math><?CDATA $(u_\star,0)$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn14.gif" xlink:type="simple" /></jats:inline-formula>of (<jats:inline-formula><jats:tex-math><?CDATA $\star$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mo>⋆</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn15.gif" xlink:type="simple" /></jats:inline-formula>) is stable with respect to a suitably chosen norm topology.</jats:p>