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• W3083882383 abstract "Abstract The chemotaxis-growth system ($star$) <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=0pt displaystyle=true rowspacing=0pt> <m:mtr> <m:mtd columnalign=right> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:mtd> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mo>∇</m:mo> </m:mrow> <m:mo>⋅</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=right> <m:msub> <m:mi>v</m:mi> <m:mi>t</m:mi> </m:msub> </m:mtd> <m:mtd columnalign=left> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>κ</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> {}left{begin{aligned} displaystyle{}u_{t}&displaystyle=DDelta u-chi% nablacdot(unabla v)+rho u-mu u^{alpha}, displaystyle v_{t}&displaystyle=dDelta v-kappa v+lambda uend{aligned}right. is considered under homogeneous Neumann boundary conditions in smoothly bounded domains <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {Omegasubsetmathbb{R}^{n}} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {ngeq 1} . For any choice of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>α</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {alpha>1} , the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($star$), the present work shows that, whenever <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>α</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>-</m:mo> <m:mfrac> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mfrac> </m:mrow> </m:mrow> </m:math> {alphageq 2-frac{2}{n}} , under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo maxsize=120% minsize=120%>(</m:mo> <m:msup> <m:mrow> <m:mo maxsize=120% minsize=120%>(</m:mo> <m:mfrac> <m:mi>ρ</m:mi> <m:mi>μ</m:mi> </m:mfrac> <m:mo maxsize=120% minsize=120%>)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:msup> <m:mo>,</m:mo> <m:mrow> <m:mfrac> <m:mi>λ</m:mi> <m:mi>κ</m:mi> </m:mfrac> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo maxsize=120% minsize=120%>(</m:mo> <m:mfrac> <m:mi>ρ</m:mi> <m:mi>μ</m:mi> </m:mfrac> <m:mo maxsize=120% minsize=120%>)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:msup> </m:mrow> <m:mo maxsize=120% minsize=120%>)</m:mo> </m:mrow> </m:math> {bigl{(}bigl{(}frac{rho}{mu}bigr{)}^{frac{1}{alpha-1}},frac{lambda}{% kappa}bigl{(}frac{rho}{mu}bigr{)}^{frac{1}{alpha-1}}bigr{)}} in the large time limit." @default.
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• W3083882383 date "2020-09-03" @default.
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• W3083882383 title "Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions" @default.
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