Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions
Michael Winkler
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=D\Delta u-\chi% \nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&\displaystyle=d\Delta v-\kappa v+\lambda u\end{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> {\Omega\subset\mathbb{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> {n\geq 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}