<p>This paper is concerned with the following quasilinear chemotaxis-consumption system <Equation ID="Equ49"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot (\nabla u^m - uv \nabla v) + au - bu^2, & (x, t) \in \Omega \times (0, \infty ), \\ v_t = \Delta v - uv, & (x, t) \in \Omega \times (0, \infty ), \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <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 stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <msup> <mi>u</mi> <mi>m</mi> </msup> <mo>-</mo> <mi>u</mi> <mi>v</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mi>u</mi> <mo>-</mo> <mi>b</mi> <msup> <mi>u</mi> <mn>2</mn> </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> <mi>u</mi> <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> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under the homogeneous Neumann boundary condition in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n\,\,(n\ge 1)$</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> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with a smooth boundary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\partial {\Omega }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. It is shown that when <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> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are constants, then the above system admits at least one global weak solution.</p>

Global boundedness of solutions in a quasilinear chemotaxis-consumption system with degenerate signal-dependent motility

期刊论文
发表日期：

2025年11月9日

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Chun Wu

摘要

This paper is concerned with the following quasilinear chemotaxis-consumption system $\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot (\nabla u^m - uv \nabla v) + au - bu^2, & (x, t) \in \Omega \times (0, \infty ), \\ v_t = \Delta v - uv, & (x, t) \in \Omega \times (0, \infty ), \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = \nabla \cdot (\nabla u^{m} - u v \nabla v) + a u - b u^{2}, & (x, t) \in Ω \times (0, \infty), \\ v_{t} = Δ v - u v, & (x, t) \in Ω \times (0, \infty), \end{matrix}) \end{matrix}$ under the homogeneous Neumann boundary condition in $\Omega \subset \mathbb {R}^n\,\,(n\ge 1)$ $Ω \subset R^{n} (n \geq 1)$ with a smooth boundary $\partial {\Omega }$ $\partial Ω$ . It is shown that when $a,b>0$ $a, b > 0$ and $m>1$ $m > 1$ are constants, then the above system admits at least one global weak solution.