A degenerate migration-consumption model in domains of arbitrary dimension

In a smoothly bounded convex domain <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> Ω⊂ Rⁿ with n ≥ 1, a no-flux initial-boundary value problem for <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> casesₜ=Δ(uφ (v)), \\ vₜ=Δv-uv, cases is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by <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> φ (ξ )=ξ α, ξ ∈ [0,ξ ₀]. By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy <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><</m:mo> <m:mi>∞</m:mi> <m:mspace width="2em"/> <m:mtext>for all </m:mtext> <m:mi>T</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> C(T):=t∈ (0,T)ess\,sup∫ _Ωu(· ,t)lnu(· ,t)< ∞ for\,all\,T >0, with sup T >0 C ( T ) < ∞ if α ∈ [1, 2].