<p>This paper deals with the following system with signal-dependent motility <Equation ID="Equ52"> <EquationSource Format="TEX">${\left\{ \begin{array}{ll} u_t = \nabla \cdot \big ( \phi (v) \nabla u - u \varphi (v) \nabla v \big ) + au - bu^l, & (x,t) \in \Omega \times (0,\infty ), \\ v_t = \Delta v - u^\gamma v, & (x,t) \in \Omega \times (0,\infty ), \end{array}\right. }$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>-</mo> <mi>u</mi> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mi>u</mi> <mo>-</mo> <mi>b</mi> <msup> <mi>u</mi> <mi>l</mi> </msup> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <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>∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <msup> <mi>u</mi> <mi>γ</mi> </msup> <mi>v</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <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>∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>under homogeneous Neumann boundary conditions in a smooth bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). Here, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a,b > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$l>2,\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>></mo> <mn>2</mn> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{l}{\gamma }>\frac{n+2}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>l</mi> <mi>γ</mi> </mfrac> <mo>></mo> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the function <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\phi \in C^2([0,\infty ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\phi (s)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$s\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varphi (s)=(\alpha - 1)\phi '(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then the considered system possesses a global classical solutions which are uniformly bounded.</p>

Global Boundedness of a Chemotaxis System with Signal-Dependent Motility and Signal Consumption

期刊论文
发表日期：

2026年2月26日

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Chun Wu

摘要

This paper deals with the following system with signal-dependent motility ${\left\{ \begin{array}{ll} u_t = \nabla \cdot \big ( \phi (v) \nabla u - u \varphi (v) \nabla v \big ) + au - bu^l, & (x,t) \in \Omega \times (0,\infty ), \\ v_t = \Delta v - u^\gamma v, & (x,t) \in \Omega \times (0,\infty ), \end{array}\right. }$ $\{\begin{matrix} u_{t} = \nabla \cdot (ϕ (v) \nabla u - u φ (v) \nabla v) + a u - b u^{l}, & (x, t) \in Ω \times (0, \infty), \\ v_{t} = Δ v - u^{γ} v, & (x, t) \in Ω \times (0, \infty), \end{matrix})$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ ( $n\ge 2$ $n \geq 2$ ). Here, $a,b > 0$ $a, b > 0$ , $l>2,\gamma >0$ $l > 2, γ > 0$ and $\frac{l}{\gamma }>\frac{n+2}{2}$ $\frac{l}{γ} > \frac{n + 2}{2}$ , the function $\phi \in C^2([0,\infty ))$ $ϕ \in C^{2} ([0, \infty))$ satisfies $\phi (s)>0$ $ϕ (s) > 0$ for all $s\ge 0$ $s \geq 0$ , and $\varphi (s)=(\alpha - 1)\phi '(s)$ $φ (s) = (α - 1) ϕ^{'} (s)$ with $\alpha \in (0,1)$ $α \in (0, 1)$ , then the considered system possesses a global classical solutions which are uniformly bounded.