Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow

Abstract

This work is concerned with the doubly degenerate cross-diffusion system <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals left-parenthesis u v u Subscript x Baseline right-parenthesis Subscript x Baseline minus left-parenthesis u squared v v Subscript x Baseline right-parenthesis Subscript x Baseline plus u v comma 2nd Row v Subscript t Baseline equals v Subscript x x Baseline minus u v comma EndLayout EndLayout"> <mml:semantics> <mml:mtable side="left" displaystyle="false"> <mml:mlabeledtr> <mml:mtd> <mml:mtext>(*)</mml:mtext> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left" rowspacing="0.683em 0.4em" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mi>v</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>v</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:mi>v</mml:mi> <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:msub> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <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:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> </mml:mtd> </mml:mlabeledtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{equation}\tag {*} \left \{ \begin {array}{l} u_t = (uvu_x)_x - (u^2 vv_x)_x + uv, \\[1mm] v_t = v_{xx}-uv, \end{array} \right . \end{equation}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> that has been proposed as a model for experimentally observable quite complex pattern formation phenomena in bacterial populations. It is shown that for any initial data satisfying adequate regularity and positivity assumptions, a no-flux initial-boundary value problem for the above in a bounded real interval possesses a global weak solution which is continuous in its first and essentially smooth in its second component. This solution is seen to asymptotically stabilize in the sense that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk asterisk right-parenthesis EndLabel u left-parenthesis dot comma t right-parenthesis right-arrow u Subscript normal infinity Baseline and v left-parenthesis dot comma t right-parenthesis right-arrow 0 as t right-arrow normal infinity EndLayout"> <mml:semantics> <mml:mtable side="left" displaystyle="false"> <mml:mlabeledtr> <mml:mtd> <mml:mtext>(**)</mml:mtext> </mml:mtd> <mml:mtd> <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:msub> <mml:mi>u</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mspace width="1em"/> <mml:mtext>and</mml:mtext> <mml:mspace width="1em"/> <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:mspace width="2em"/> <mml:mtext>as </mml:mtext> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mtd> </mml:mlabeledtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{equation}\tag {**} u(\cdot ,t) \to u_\infty \quad \text {and} \quad v(\cdot ,t)\to 0 \qquad \text {as } t\to \infty \end{equation}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with some nonnegative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript normal infinity Baseline element-of upper C Superscript 0 Baseline left-parenthesis normal upper

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