具有密度抑制运动性、θ-逻辑源和间接信号产生且在强阿利效应下的趋化系统解的整体存在性与唯一性

中文摘要 AI 辅助

我们考虑如下趋化-生长系统，其具有密度抑制运动性和包含强阿利效应的θ-逻辑增长项：\n$$\n\left\{ \begin{aligned} u_t &= \Delta(\gamma(v)u) + \mu u(1-u^\theta)(u-A), & x \in \Omega, \quad t > 0, \\\\ v_t &= \Delta v - v+ w^\beta, & x \in \Omega,\quad t > 0, \\\\ w_t &= -\delta w +u, & x \in \Omega, \quad t > 0, \end{aligned} \right.\n$$\n该系统服从有界区域$\Omega\subset \mathbb{R}^d (d\geq 2)$上的齐次诺伊曼边界条件，区域具有光滑边界。其中$\mu\in \mathbb{R}, \delta,\beta>0, \theta\geq1$，正运动性函数$\gamma(v) \in C^3([0,\infty))$满足对所有$v\geq 0$有$\gamma'(v) \leq 0$。本文的主要目标是建立所提问题经典解的整体存在性与有界性。我们的分析结合了用于证明局部时间存在性的Schauder不动点定理、可延拓性准则、基础能量估计、$L^p$界、Gagliardo-Nirenberg插值不等式、Young不等式、半群估计以及Moser型迭代格式，以推导时间一致的$L^\infty$界，从而确保整体有界经典解。更确切地说，我们证明当$\beta < \frac{2(\theta+2)}{d}$时，该趋化-生长系统存在唯一的整体有界经典解。此外，若假设运动性函数的对数导数在$[0,\infty)$上一致有界，即$\frac{\gamma'(v)}{\gamma(v)} \in L^{\infty}([0,\infty))$，则当$\theta > \frac{4-d}{d-2}$时，$\beta$的上述限制可放宽至$\beta < \frac{d(\theta+1)+2(\theta+2)}{2d}$。

英文摘要

We consider the following chemotaxis-growth system, featuring density-suppressed motility and a $θ$-logistic growth term that incorporates a strong Allee effect: \begin{equation*} \left\{ \begin{aligned} u_t &= Δ(γ(v)u) + μu(1-u^θ)(u-A), & x \in Ω, \quad t > 0, v_t &= Δv - v+ w^β, & x \in Ω,\quad t > 0, w_t &= -δw +u, & x \in Ω, \quad t > 0, \end{aligned} \right. \end{equation*} subject to homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 2)$ with smooth boundary. Here $μ\in \mathbb{R} , δ,β>0, θ\geq1$, and the positive motility function $γ(v) \in C^3([0,\infty))$ fulfills $γ'(v) \leq 0$ for all $v\geq 0$. The main objective of this paper is to establish the global existence and boundedness of classical solutions to the proposed problem. Our analysis combines the Schauder fixed point theorem for proving local-in-time existence with the extensibility criterion, fundamental energy estimates, $L^p-$bounds, the Gagliardo-Nirenberg interpolation inequality, Young's inequality, semigroup estimates, and a Moser-type iteration scheme to derive uniform-in-time $L^{\infty}-$bounds, thereby ensuring global bounded classical solutions. More precisely, we establish that the chemotaxis-growth system admits a unique globally bounded classical solution whenever $β< \frac{2(θ+2)}{d}$. Furthermore, if the logarithmic derivative of the motility function is assumed to be uniformly bounded on $[0,\infty)$, that is, $\frac{γ'(v)}{γ(v)} \in L^{\infty}([0,\infty))$, then the above restriction on $β$ can be weakened to $β< \frac{d(θ+1)+2(θ+2)}{2d}$, provided that $θ> \frac{4-d}{d-2}$.

发表机构

• Indian Institute of Technology Roorkee（印度理工学院鲁尔基分校）

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