论文标题
具有异质逻辑资源的凯勒 - 塞格系统何时会允许通用解决方案？
When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?
论文作者
论文摘要
我们为趋化系统构建全球通用解决方案 \ begin {align*} \ begin {case} u_t =ΔU-\ nabla \ cdot（u \ nabla v） +λ（x）u -μ（x）u^κ，\\ v_t =ΔV -V + U \ end {cases} \ end {align*}在平滑的，有限的域中$ω\ subset \ mathbb r^n $，$ n \ geq 2 $，对于$λ，μ$和$κ$的某些选择。 在这里，除此之外，选择$μ（x）= | x |^α$带有$α<2 $和$κ= 2 $以及$μ\equivμ_1> 0 $和$κ> \ min \ min \ min \ min \ {\ frac {\ frac {2n-2}足够光滑的$λ$）。 尽管前一种情况似乎一般是新颖的，但在两维环境中，后者在Winkler（Adv。NonlinearAnal。9（2019），第1、526-566号）的最新结果上有所改善，其中条件$κ> \ frac {2n+4} {N+4} {N+4} $已阻碍。特别是，对于$ n = 2 $，我们的结果表明，要服用任何$κ> 1 $，就足以排除倒塌成持续的迪拉克分布的可能性。
We construct global generalized solutions to the chemotaxis system \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λ(x) u - μ(x) u^κ,\\ v_t = Δv - v + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \geq 2$, for certain choices of $λ, μ$ and $κ$. Here, inter alia, the selections $μ(x) = |x|^α$ with $α< 2$ and $κ= 2$as well as $μ\equiv μ_1 > 0$ and $κ> \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\}$ are admissible (in both cases for any sufficiently smooth $λ$). While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1, 526-566), where the condition $κ> \frac{2n+4}{n+4}$ has been imposed. In particular, for $n = 2$, our result shows that taking any $κ> 1$ suffices to exclude the possibility of collapse into a persistent Dirac distribution.
