<p>In this paper, we consider the fully parabolic nutrient taxis system <Equation ID="Equ55"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2555_Article_Equ55.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="399" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{lll} u_t=d_1\Delta u^m-\nabla \cdot (u\chi (v)\nabla v), & x\in \Omega , t>0, \\ v_t=d_2\Delta v-\xi ug(v)-\mu v+r(x,t), & x\in \Omega , t>0 \end{array}\right. () \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <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> <mi>u</mi> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <mi>ξ</mi> <mi>u</mi> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>μ</mi> <mi>v</mi> <mo>+</mo> <mi>r</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mrow> <mo stretchy="false">(</mo> <mrow /> <mo>∗</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. We show that if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2555_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$m>\frac{3}{2}-\frac{1}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the system (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2555_Article_IEq2.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>) possesses at least one global bounded weak solution in high-dimensional domain.</p>

Global existence of weak solutions to a nutrient taxis model with porous medium diffusion

期刊论文
发表日期：

2025年8月6日

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Linsong Wang,
Yanxi Li,
Guoqiang Ren

摘要

In this paper, we consider the fully parabolic nutrient taxis system $\begin{aligned} \left\{ \begin{array}{lll} u_t=d_1\Delta u^m-\nabla \cdot (u\chi (v)\nabla v), & x\in \Omega , t>0, \\ v_t=d_2\Delta v-\xi ug(v)-\mu v+r(x,t), & x\in \Omega , t>0 \end{array}\right. (*) \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = d_{1} Δ u^{m} - \nabla \cdot (u χ (v) \nabla v), & x \in Ω, t > 0, \\ v_{t} = d_{2} Δ v - ξ u g (v) - μ v + r (x, t), & x \in Ω, t > 0 \end{matrix}) (*) \end{matrix}$ under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. We show that if $m>\frac{3}{2}-\frac{1}{N}$ $m > \frac{3}{2} - \frac{1}{N}$ , the system ( $*$ $*$ ) possesses at least one global bounded weak solution in high-dimensional domain.