<p>This paper investigates the following quasilinear chemotaxis system with consumption of chemoattractant <Equation ID="Equ60"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta {u^m}-\nabla \cdot (u^{q-1}\nabla {v}), & (x,t)\in \Omega \times (0,\infty ), \\ v_t=\Delta {v}-uv, & (x,t)\in \Omega \times (0,\infty ) \end{array}\right. } \end{aligned}$</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">Δ</mi> <msup> <mi>u</mi> <mi>m</mi> </msup> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <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> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under a smooth bounded convex domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n\,\,(n>2)$</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>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with 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>, where the parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m>1,~q\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. It is shown that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q>m+\frac{2}{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mi>m</mi> <mo>+</mo> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, for any sufficiently small initial data, the associated initial-boundary value problem possesses a globally bounded weak solution.</p>

Global existence of weak solutions to a quasilinear chemotaxis system with consumption of chemoattractant

期刊论文
发表日期：

2025年12月9日

查看全文

Chun Wu

摘要

This paper investigates the following quasilinear chemotaxis system with consumption of chemoattractant $\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta {u^m}-\nabla \cdot (u^{q-1}\nabla {v}), & (x,t)\in \Omega \times (0,\infty ), \\ v_t=\Delta {v}-uv, & (x,t)\in \Omega \times (0,\infty ) \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u^{m} - \nabla \cdot (u^{q - 1} \nabla v), & (x, t) \in Ω \times (0, \infty), \\ v_{t} = Δ v - u v, & (x, t) \in Ω \times (0, \infty) \end{matrix}) \end{matrix}$ under a smooth bounded convex domain $\Omega \subset \mathbb {R}^n\,\,(n>2)$ $Ω \subset R^{n} (n > 2)$ with smooth boundary $\partial {\Omega }$ $\partial Ω$ , where the parameters $m>1,~q\ge 2$ $m > 1, q \geq 2$ . It is shown that if $q>m+\frac{2}{n}$ $q > m + \frac{2}{n}$ , for any sufficiently small initial data, the associated initial-boundary value problem possesses a globally bounded weak solution.