<p>In this paper, we study a chemotaxis system with indirect signal production/consumption and signal-dependent motility under homogeneous Neumann boundary conditions in a smoothly 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>, which described as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_t=\Delta (\gamma (v)u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$v_t=d_v\Delta v+f(v,w)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>d</mi> <mi>v</mi> </msub> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$w_t=d_w\Delta w-w+u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>d</mi> <mi>w</mi> </msub> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>-</mo> <mi>w</mi> <mo>+</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(v,w)=-v+w$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>-</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation> and the motility function satisfies <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma \in C^3((0,\infty ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>3</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>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma (s)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$s>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we proved that the system admits a global classical solution for any appropriately regular initial value when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and further, if we exclude the singular at <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$s=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, then the global smooth solution is bounded-in-time. While if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$f(v,w)=-vw$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>-</mo> <mi>v</mi> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma \in C^3([0,\infty ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>3</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>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we showed the existence and asymptotic behavior of globally bounded solutions when either <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n\le 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Vert v_0\Vert _{L^\infty (\Omega )}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is suitably small.</p>

Global existence and boundedness in a chemotaxis system involving indirect production/consumption mechanism and signal-dependent motility

期刊论文
发表日期：

2025年10月11日

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Quanyong Zhao,
Jinrong Wang

摘要

In this paper, we study a chemotaxis system with indirect signal production/consumption and signal-dependent motility under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ , which described as $u_t=\Delta (\gamma (v)u)$ $u_{t} = Δ (γ (v) u)$ , $v_t=d_v\Delta v+f(v,w)$ $v_{t} = d_{v} Δ v + f (v, w)$ , $w_t=d_w\Delta w-w+u$ $w_{t} = d_{w} Δ w - w + u$ . If $f(v,w)=-v+w$ $f (v, w) = - v + w$ and the motility function satisfies $\gamma \in C^3((0,\infty ))$ $γ \in C^{3} ((0, \infty))$ , $\gamma (s)>0$ $γ (s) > 0$ for $s>0$ $s > 0$ , we proved that the system admits a global classical solution for any appropriately regular initial value when $n\le 3$ $n \leq 3$ , and further, if we exclude the singular at $s=0$ $s = 0$ , then the global smooth solution is bounded-in-time. While if $f(v,w)=-vw$ $f (v, w) = - v w$ and $\gamma \in C^3([0,\infty ))$ $γ \in C^{3} ([0, \infty))$ , $\gamma >0$ $γ > 0$ on $[0,\infty )$ $[0, \infty)$ , we showed the existence and asymptotic behavior of globally bounded solutions when either $n\le 5$ $n \leq 5$ or $\Vert v_0\Vert _{L^\infty (\Omega )}$ $‖ v_{0} ‖_{L^{\infty} (Ω)}$ is suitably small.