论文标题
具有空间异质逻辑源的趋化系统中爆炸的可能点
Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source
论文作者
论文摘要
我们讨论了可能的空间不均匀性在逻辑源术语系数中的影响，形式\ begin的抛物线 - 纤维化趋化性增长系统\ begin {align*} u_t*} u_t＆=ΔU-\ nabla \ nabla \ cdot \ cdot（u \ nabla v）v）v）v） +κ（x）在平稳的域中，$ω\ subset \ mathbb {r}^2 $。假设系数功能满足$κ，μ\ in c^0（\overlineΩ）$带有$μ\ geq0 $，我们证明，经典解决方案的有限时间爆炸只能发生在$μ$中的点零，即blight-mathcal nebcal {b} $ innign* \ big \ {x \ in \overlineΩ\midμ（x）= 0 \ big \}。此外，我们表明，每当$μ（x_0）> 0 $对于某些$ x_0 \ in \overlineΩ$中的$ 0 $，那么人们就可以找到一个开放的邻居$ u $ u $ $ x_0 $ in $ \overlineΩ$，这样$ u $ u $ u $在整个进化过程中都保持在$ u $中。
We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth systems of the form \begin{align*} u_t &= Δu - \nabla\cdot(u\nabla v) + κ(x)u-μ(x)u^2, 0 &= Δv - v + u \end{align*} in smoothly bounded domains $Ω\subset\mathbb{R}^2$. Assuming that the coefficient functions satisfy $κ,μ\in C^0(\overlineΩ)$ with $μ\geq0$ we prove that finite-time blow-up of the classical solution can only occur in points where $μ$ is zero, i.e.\ that the blow-up set $\mathcal{B}$ is contained in \begin{align*} \big\{x\in\overlineΩ\midμ(x)=0\big\}. \end{align*} Moreover, we show that whenever $μ(x_0)>0$ for some $x_0\in\overlineΩ$, then one can find an open neighbourhood $U$ of $x_0$ in $\overlineΩ$ such that $u$ remains bounded in $U$ throughout evolution.
