<p>In this paper, we consider a nonlinear chemotaxis system with indirect signal consumption <Equation ID="Equ69"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u^m-\nabla \cdot \big (u\nabla v\big )+\mu u(1-u),&\quad x\in \Omega ,\,t>0,\\&v_t=\Delta v-vw,&\quad x\in \Omega ,\,t>0,\\&w_t=-\delta w+u,&\quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>u</mi> <mi>m</mi> </msup> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>u</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>μ</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <mi>v</mi> <mi>w</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>δ</mi> <mi>w</mi> <mo>+</mo> <mi>u</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under homogeneous Neumann boundary conditions in a smooth bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the problem possesses a global weak solution, which is uniformly bounded.</p>

Boundedness and global solvability for a three-dimensional nonlinear chemotaxis system with indirect signal consumption

期刊论文
发表日期：

2026年5月24日

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Ju Zhou

摘要

In this paper, we consider a nonlinear chemotaxis system with indirect signal consumption $\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u^m-\nabla \cdot \big (u\nabla v\big )+\mu u(1-u),&\quad x\in \Omega ,\,t>0,\\&v_t=\Delta v-vw,&\quad x\in \Omega ,\,t>0,\\&w_t=-\delta w+u,&\quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u^{m} - \nabla \cdot (u \nabla v) + μ u (1 - u), & x \in Ω, t > 0, \\ v_{t} = Δ v - v w, & x \in Ω, t > 0, \\ w_{t} = - δ w + u, & x \in Ω, t > 0 \end{matrix}) \end{matrix}$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ , where $\delta >0$ $δ > 0$ , $\mu >0$ $μ > 0$ and $m>0$ $m > 0$ . For any $m>1$ $m > 1$ , we show that the problem possesses a global weak solution, which is uniformly bounded.