论文标题
具有加权反应的反应扩散方程的爆炸模式在一般维度
Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension
论文作者
论文摘要
我们将自相似形式的所有爆炸解决方案分类为以下反应 - 扩散方程$$ \ partial_tu =δu=δu^m+| x |^s |^σu^p，$$ $（x，x，x，x，t）\ in \ real^n \ time^n \ times（0，t）$ $ -2（P-1）/（M-1）<σ<\ infty $。我们证明，相对于原点附近的局部行为有几种类型的自相似解，它们的存在取决于$σ$的大小。特别是，这些解决方案具有不同的爆破集和费率：其中有些具有$ x = 0 $作为爆炸点，其他一些唯一的唯一在（太空）无穷大。因此，我们强调重量对方程式爆破模式的特定形式的影响。本研究概括了作者的先前作品，仅限于维度$ n = 1 $和$σ> 0 $。
We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p<m$ and $-2(p-1)/(m-1)<σ<\infty$. We prove that there are several types of self-similar solutions with respect to the local behavior near the origin, and their existence depends on the magnitude of $σ$. In particular, these solutions have different blow-up sets and rates: some of them have $x=0$ as a blow-up point, some other only blow up at (space) infinity. We thus emphasize on the effect of the weight on the specific form of the blow-up patterns of the equation. The present study generalizes previous works by the authors limited to dimension $N=1$ and $σ>0$.
