论文标题
双曲线空间锥中半线性抛物线方程的爆破
Blow-up for semilinear parabolic equations in cones of the hyperbolic space
论文作者
论文摘要
我们研究了时间非全球在半线性热方程式中的全球溶液的存在和不存在，其反应项是$ e^{μt} u^p $（$μ\ in \ mathbb {r}，p> 1 $），在高度骨空间的圆锥体上构成。在$μ$和$ p $的某个假设下，与$-Δ$ in $ \ m m mathbb h^n $的底部有关，我们证明，对于任何非平凡的非负初始基准，任何解决方案都会在有限的时间内爆炸。取而代之的是，如果参数$μ$和$ p $满足我们所拥有的相反条件：（a）当初始基准足够大时爆炸，（b）当初始数据足够小时，存在全局解决方案。因此，我们在参数上的条件$μ$和$ p $是最佳的。 我们看到爆炸和全球存在并不取决于锥体的幅度。这与欧几里得环境中发生的情况大不相同，这实际上是由于$ \ Mathbb H^n $的特定几何特征。
We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{μt}u^p$ ($μ\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $μ$ and $p$, related to the bottom of the spectrum of $-Δ$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $μ$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters $μ$ and $p$ are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting, and it is essentially due to a specific geometric feature of $\mathbb H^n$.
