论文标题
在一般曲率边界下的cartan-hadamard歧管上具有大数据的多孔培养基方程
The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds
论文作者
论文摘要
我们考虑了Caran-Hadamard歧管上多孔培养基方程的Cauchy问题的非常弱的解决方案，这些方程式被认为可以满足一般的曲率界限并随机完成。我们确定可以以规定的速率以无穷大生长的初始数据，这取决于通过积分函数假定的曲率界限，因此，相应的解决方案至少在$ [0，t] $上存在，适用于$ t> 0 $。根据初始基准的合适加权规范估算了最大的存在时间$ t $。我们的结果很敏锐，从某种意义上说，增长率较慢产生了全球存在，而人们可以构建具有关键增长的数据，相应的解决方案在有限的时间内爆炸。在进一步的假设下，在同一增长类别中也证明了非常弱的解决方案的唯一性。
We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on $[0,T]$ for a suitable $T>0$. The maximal existence time $T$ is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class.
