<p>In this paper, we consider the following consumption chemotaxis system <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_Equa.gif" Format="GIF" Height="85" Rendition="HTML" Resolution="72" Type="Linedraw" Width="366" /> </MediaObject> <EquationSource Format="MATHML"><math> <mrow> <mo>{</mo> <mtable> <mtr> <mtd columnalign="left"> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo>(</mo> <mi>u</mi> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mi>u</mi> <mo>−</mo> <mi>b</mi> <msup> <mi>u</mi> <mi>γ</mi> </msup> <mo>,</mo> </mtd> <mtd columnalign="left"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mi>v</mi> <mi>w</mi> <mo>,</mo> </mtd> <mtd columnalign="left"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <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> </mtd> <mtd columnalign="left"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> <EquationSource Format="TEX">$ \textstyle\begin{cases} u_{t}=\Delta \left (u\phi (v)\right )+au-bu^{\gamma }, &(x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta {v}-uvw, & (x,t)\in \Omega \times (0,\infty ), \\ w_{t}=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{cases} $</EquationSource> </Equation> under the smooth bounded domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mspace width="0.2em" /> <mspace width="0.2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset \mathbb{R}^{n}\,\,(n\ge 2)$</EquationSource> </InlineEquation> with homogeneous Neumann boundary conditions, where the parameters <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>a</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$a>0$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>b</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$b>0$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>γ</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$\gamma \ge 2$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_738_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>δ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\delta >0$</EquationSource> </InlineEquation>. It has been shown that for any sufficiently regular initial data, the associated initial-boundary value problem has a global classical solutions.</p>

Global Boundedness in a Chemotaxis System Involving Signal-Dependent Motility and Indirect Signal Consumption

期刊论文
发表日期：

2025年8月25日

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Chun Wu

摘要

In this paper, we consider the following consumption chemotaxis system ${\begin{matrix} u_{t} = Δ (u ϕ (v)) + a u - b u^{γ}, & (x, t) \in Ω \times (0, \infty), \\ v_{t} = Δ v - u v w, & (x, t) \in Ω \times (0, \infty), \\ w_{t} = - δ w + u, & (x, t) \in Ω \times (0, \infty), \end{matrix}$ $ \textstyle\begin{cases} u_{t}=\Delta \left (u\phi (v)\right )+au-bu^{\gamma }, &(x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta {v}-uvw, & (x,t)\in \Omega \times (0,\infty ), \\ w_{t}=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{cases} $ under the smooth bounded domain $Ω \subset R^{n} (n \geq 2)$ $\Omega \subset \mathbb{R}^{n}\,\,(n\ge 2)$ with homogeneous Neumann boundary conditions, where the parameters $a > 0$ $a>0$ , $b > 0$ $b>0$ , $γ \geq 2$ $\gamma \ge 2$ and $δ > 0$ $\delta >0$ . It has been shown that for any sufficiently regular initial data, the associated initial-boundary value problem has a global classical solutions.