Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

<jats:title>Abstract</jats:title> <jats:p>The chemotaxis-growth system</jats:p> <jats:p> <jats:disp-formula id="j_ans-2020-2107_eq_0001"> <jats:label>($\star$)</jats:label> <jats:alternatives> <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> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_fig_001.png" /> <jats:tex-math>{}\left\{\begin{aligned} \displaystyle{}u_{t}&amp;\displaystyle=D\Delta u-\chi% \nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&amp;\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right.</jats:tex-math> </jats:alternatives> </jats:disp-formula> </jats:p> <jats:p>is considered under homogeneous Neumann boundary conditions in smoothly bounded domains <jats:inline-formula id="j_ans-2020-2107_ineq_9999"> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_inl_001.png" /> <jats:tex-math>{\Omega\subset\mathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula id="j_ans-2020-2107_ineq_9998"> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_inl_002.png" /> <jats:tex-math>{n\geq 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. For any choice of <jats:inline-formula id="j_ans-2020-2107_ineq_9997"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_inl_003.png" /> <jats:tex-math>{\alpha&gt;1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, 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 <jats:inline-formula id="j_ans-2020-2107_ineq_9996"> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_inl_004.png" /> <jats:tex-math>{\alpha\geq 2-\frac{2}{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state <jats:inline-formula id="j_ans-2020-2107_ineq_9995"> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2107_inl_005.png" /> <jats:tex-math>{\bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{% \kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in the large time limit.</jats:p>