论文标题
具有对数灵敏度的甲状双曲线凯勒 - 塞格乳清系统的有限时间爆炸
Finite-time blow-up to hyperbolic Keller-Segel system of consumption type with logarithmic sensitivity
论文作者
论文摘要
本文使用对数灵敏度\ begin {equation*}涉及双曲线凯勒 - 塞格的消费类型系统 \ partial_ {t}ρ= - χ\ nabla \ cdot \ left（ρ\ nabla \ log c \ right），\ quad \ partial_ {t} c = - μcρ\ quad（χ，\ quad（χ，\，\，μ> 0） （d \ ge1）$用于非变化的初始数据。该系统与肿瘤血管生成密切相关，肿瘤血管生成是趋化性的重要例子。我们首先显示了对应于非消失平滑初始数据的平滑解决方案的局部存在。接下来，通过Riemann Infortiants，我们在$ d = 1 $时提供了此初始数据的一些足够条件，以进行有限的时间奇异性形成。然后，我们证明，对于任何$ d \ ge1 $，在有限的时间内，某些不变的$ c^\ infty $ -data可能会变得奇异。此外，我们得出有关奇点发生时解决方案行为的详细信息。特别是，此信息表明，来自某些初始数据的奇异性形成不是因为$ c $触摸零（这使$ \ log c $ diverge），而是由于$ c^1 \ times c^1 \ times c^2 $ -norm的爆炸，$（ρ，c）$。作为推论，我们还构建了在任何恒定平衡状态附近的初始数据，该数据在任何$ d \ ge1 $中都在有限的时间内炸毁。我们的结果是在\ cite {ij21}中的有限时间爆破的扩展，其中需要初始数据以满足某些消失的条件。此外，我们以某种方式来解释我们的结果，即某种类型的阻尼或耗散$ρ$必须必须确保全球平滑解决方案的存在，即使初始数据在恒定平衡状态周围很小。
This paper deals with a hyperbolic Keller-Segel system of consumption type with the logarithmic sensitivity \begin{equation*} \partial_{t} ρ= - χ\nabla \cdot \left (ρ\nabla \log c\right),\quad \partial_{t} c = - μcρ\quad (χ,\,μ>0) \end{equation*} in $\mathbb{R}^d\; (d \ge1)$ for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. We firstly show the local existence of smooth solutions corresponding to nonvanishing smooth initial data. Next, through Riemann invariants, we present some sufficient conditions of this initial data for finite-time singularity formation when $d=1$. We then prove that for any $d\ge1$, some nonvanishing $C^\infty$-data can become singular in finite time. Moreover, we derive detailed information about the behaviors of solutions when the singularity occurs. In particular, this information tells that singularity formation from some initial data is not because $c$ touches zero (which makes $\log c$ diverge) but due to the blowup of $C^1\times C^2$-norm of $(ρ,c)$. As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any $d\ge1$. Our results are the extension of finite-time blow-up results in \cite{IJ21}, where initial data is required to satisfy some vanishing conditions. Furthermore, we interpret our results in a way that some kinds of damping or dissipation of $ρ$ are necessarily required to ensure the global existence of smooth solutions even though initial data are small perturbations around constant equilibrium states.
