论文标题
具有强度强的反应扩散方程的自相似爆炸曲线
Self-similar blow-up profiles for a reaction-diffusion equation with strong weighted reaction
论文作者
论文摘要
我们研究与以下二阶反应扩散方程相关的自相似爆炸曲线，并具有强大的加权反应和无界的重量：$$ \ partial_tu = \ partial_ {xx}（xx}（u^m） + | x |^s |^σu^p，$$ $ x \ $ x $σ> 2（1-P）/（M-1）$。作为第一个结果，我们表明，有限的时间爆破解决方案以$ m+p> 2 $> 2 $和$σ$在被考虑的范围内存在，这一事实是全新的：在已经研究的没有重量的已经研究的反应 - 扩散方程中，当$ p <1 $的情况下，没有有限的时间爆炸。此外，我们证明，如果满足条件$ m+p> 2 $，则所有相似的爆炸配置文件都得到紧凑的支持，并且存在\ emph {两个不同的接口行为}用于方程解决方案的解决方案，对应于两个不同的接口方程。我们对具有两种类型接口的自相似爆炸曲线进行了分类，并表明在某些情况下会发生\ Emph {全局爆破}，在某些情况下，有限的时间爆破发生\ Emph {仅在空间Infinity}。我们还表明，如果$ m+p <2 $，没有自相似的解决方案，而关键范围$ m+p = 2 $，$σ> 2 $由于显着的技术差异而被推迟到其他工作中。
We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ and $σ>2(1-p)/(m-1)$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and $σ$ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the condition $m+p>2$ is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist \emph{two different interface behaviors} for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases \emph{global blow-up} occurs, and in some other cases finite time blow-up occurs \emph{only at space infinity}. We also show that there is no self-similar solution if $m+p<2$, while the critical range $m+p=2$ with $σ>2$ is postponed to a different work due to significant technical differences.
