论文标题
反应扩散方程的自相似爆炸曲线具有强烈的加权反应
Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction
论文作者
论文摘要
我们对以下反应扩散方程的自相似爆炸曲线进行了分类，并具有关键的强度加权反应和无限重量的重量：$$ \ partial_tu = \ partial_ {xx} {xx}（u^m） + | x |^s |^σu^p，$ $ $ $ $ $ $ $σ> 2 $完成在最近的一项工作中进行的分析，在该工作中，此非常有趣的关键案例被搁置了。我们表明，以$σ> 2 $的形式存在有限的时间爆破解决方案。此外，所有的爆炸配置文件都具有紧凑的支持，它们的支持为\ emph {本地化}：显式$η> 0 $，使得任何爆破配置文件都满足$ {\ rm supp} \，f \ subseteq [0，η] $。该属性是出乎意料的，与$ m+p> 2 $的范围形成鲜明对比。我们还对原点附近的轮廓的可能行为进行了分类。
We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ such that $m+p=2$ and $σ>2$ 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 $σ>2$. Moreover all the blow-up profiles have compact support and their supports are \emph{localized}: there exists an explicit $η>0$ such that any blow-up profile satisfies ${\rm supp}\,f\subseteq[0,η]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin.
