Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system

in smooth bounded domains \(\Omega \subset {\mathbb {R}}^n\), \(n \ge 1\), are known to be global in time if \(\lambda \ge 0\), \(\mu > 0\) and \(\kappa > 2\). In the present work, we show that the exponent \(\kappa = 2\) is actually critical in the four- and higher dimensional setting. More precisely, if

for balls \(\Omega \subset {\mathbb {R}}^n\) and parameters \(\lambda \ge 0\), \(m_0 > 0\), we construct a nonnegative initial datum \(u_0 \in C^0({{\overline{\Omega }}})\) with \(\int \nolimits _\Omega u_0 = m_0\) for which the corresponding solution (u, v) of (\(\star \)) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for \(\kappa \in (1, \frac{3}{2})\) (and \(\lambda \ge 0\), \(\mu > 0\)). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function \(w(s, t) = \int \nolimits _0^{\root n \of {s}} \rho ^{n-1} u(\rho , t) \,\mathrm {d}\rho \) fulfills the estimate \(w_s \le \frac{w}{s}\). Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, \(s_0\) and \(\gamma \), the function \(\phi (t) = \int \nolimits _0^{s_0} s^{-\gamma } (s_0 - s) w(s, t)\) cannot exist globally.

趋化系统的Neumann初边值问题的非负解

在平滑有界区域\（\欧米茄\子集{\ mathbb {R}} ^ N \） ，\（N \ GE 1 \） ，已知是全球时间如果\（\拉姆达\ GE 0 \） ，\ （\ mu> 0 \）和\（\ kappa> 2 \）。在当前的工作中，我们表明指数\（\ kappa = 2 \）实际上在四维和更高维设置中至关重要。更确切地说，如果

对于球\（\ Omega \ subset {\ mathbb {R}} ^ n \）和参数\（\ lambda \ ge 0 \），\（m_0> 0 \），我们构造了一个非负初始基准\（u_0 \ in C 1-4 0（{{\划线{\欧米茄}}}）\）与\（\ INT \ nolimits _ \欧米茄U_0 = M_0 \）的量，相应溶液（Û，  v）的（\（\星\））在限定时间内爆炸。而且，在3D中，我们获得\（\ kappa \ in（1，\ frac {3} {2}）\）（）和\（\ lambda \ ge 0 \），\（\ mu > 0 \））。作为我们分析的基石，对于某些初始数据，我们证明了质量累积函数\（w（s，t）= \ int \ nolimits _0 ^ {\ root n \ of {s}} \ rho ^ {n-1} u（\ rho，t）\，\ mathrm {d} \ rho \ ）满足估算值\（w_s \ le \ frac {w} {s} \）。使用该信息，我们然后得到有限时间吹胀的ù通过表明对于适当选择的初始数据，\（S_0 \）和\（\伽马\） ，函数\（\披（T）= \ INT \ nolimits _0 ^ {s_0} s ^ {-\ gamma}（s_0-s w（s，t）\）不能全局存在。