We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight:

posed for \(x\in {\mathbb {R}}\), \(t\ge 0\), where \(m>1\), \(0<p<1\) such that \(m+p=2\) and \(\sigma >2\) completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for \(\sigma >2\). Moreover all the blow-up profiles have compact support and their supports are localized: there exists an explicit \(\eta >0\) such that any blow-up profile satisfies \(\mathrm{supp}\,f\subseteq [0,\eta ]\). This property is unexpected and contrasting with the range \(m+p>2\). We also classify the possible behaviors of the profiles near the origin.

我们对以下具有临界强加权反应和无界重量的反应扩散方程的自相似爆炸分布进行分类：

对\（x \ in {\ mathbb {R}} \），\（t \ ge 0 \）进行定位，其中\（m> 1 \），\（0 <p <1 \）这样\（m + p = 2 \）和\（\ sigma> 2 \）完成了最近工作中进行的分析，而这一非常有趣的关键案例被搁置了。我们证明了对于\（\ sigma> 2 \）存在自相似形式的有限时间爆破解。此外，所有爆炸轮廓都具有紧凑的支撑，并且它们的支撑是局部的：存在一个明确的\（\ eta> 0 \），这样任何爆炸轮廓都可以满足\（\ mathrm {supp} \，f \ subseteq [0 ，\ eta] \）。此属性是意外的，并且与范围形成对比\（m + p> 2 \）。我们还对原点附近轮廓的可能行为进行分类。