二维双重退化营养趋肤系统中边界性研究
Boundedness in a two-dimensional doubly degenerate nutrient taxis system with logistic source
AI总结：
本文研究二维双重退化营养趋肤系统，通过加权能量方法证明了任意光滑初始数据下问题的全局弱解的均匀有界性。
摘要：
我们研究以下双重退化营养趋肤系统：u_t=∇·(u v∇u)-∇·(u²v∇v)+u-u²，v_t=Δv-u v，该系统在有界光滑域Ω⊂R²中，受齐次Neumann边界条件约束。该模型用于描述在薄琼脂板上观察到的枯草芽孢杆菌群体聚集模式。先前结果已建立一维空间中的全局有界性，并在二维情况下，在额外假设如初始数据小或凸域下，建立了全局有界性（见例如M. Winkler, Trans. Amer. Math. Soc., 2021；M. Winkler, J. Differ. Equ., 2024）。在存在二次退化项的逻辑增长中，该退化结构显著增强，通过采用加权能量方法，我们证明对于任意光滑初始数据，问题（eq-0.1）存在全局弱解，并且在时间上保持均匀有界。
英文摘要：
We are concerned with the following doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq-0.1} u_t=\nabla\cdot(u v\nabla u)-\nabla\cdot(u^{2} v\nabla v)+u-u^2,\\[1mm] v_t=Δv-u v, \end{cases} \end{align} posed in a bounded smooth domain $Ω\subset\mathbb{R}^2$ under homogeneous Neumann boundary conditions. This model was introduced to describe the aggregation patterns of colonies of \emph{Bacillus subtilis} observed on thin agar plates. Previous results have established global boundedness in one space dimension and, in two dimensions, under additional assumptions such as small initial data or convex domains (see, e.g., M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021; M. Winkler, \textit{J. Differ. Equ.}, 2024). In the presence of the quadratic degradation term in the logistic growth, which markedly enhances the dissipative structure of the system, and by employing a weighted energy method, we prove that for arbitrary smooth initial data the problem \eqref{eq-0.1} admits a global weak solution that remains uniformly bounded in time.
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