论文标题
在趋化模型中，无限时间爆炸的质量阈值
Mass threshold for infinite-time blowup in a chemotaxis model with splitted population
论文作者
论文摘要
我们研究趋化模型$ \ partial $ t u = div（$ \ nabla $ u -u $ \ u $ \ nabla $ w） + $θ$ v -u in（0，$ \ infty $）x $ \ partial $ t v = u -t v = u -$θ$ v in（0，$ \ iffty $ $ $ \ $ $ $ $ $ $ $ f = - v in（0，$ \ infty $）x $ω$，在有限且光滑的域$ω$ $ \ subset $ r 2中具有无升华边界条件，其中u和v代表了某些物种的移动和静态个体的密度，以及浓度的浓度。我们证明，在适当的功能环境中，所有解决方案都在全球范围内及时存在。此外，我们确定了整个种群u + v的临界质量m c> 0的存在，使得对于$（0，m c）中的m $ \，任何解决方案都是有限的，而对于几乎所有m> m c，在无限时间内都存在解决方案。分析的基础是Liapunov功能的构建。据我们所知，这是当大众保护包括两个亚群（不仅是移动的亚群）时，这是这种结果的第一个结果。
We study the chemotaxis model $\partial$ t u = div($\nabla$u -- u$\nabla$w) + $θ$v -- u in (0, $\infty$) x $Ω$, $\partial$ t v = u -- $θ$v in (0, $\infty$) x $Ω$, $\partial$ t w = D$Δ$w -- $α$w + v in (0, $\infty$) x $Ω$, with no-flux boundary conditions in a bounded and smooth domain $Ω$ $\subset$ R 2 , where u and v represent the densities of subpopulations of moving and static individuals of some species, respectively, and w the concentration of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence of a critical mass M c > 0 of the whole population u + v such that, for M $\in$ (0, M c), any solution is bounded, while, for almost all M > M c , there exist solutions blowing up in infinite time. The building block of the analysis is the construction of a Liapunov functional. As far as we know, this is the first result of this kind when the mass conservation includes the two subpopulations and not only the moving one.
