论文标题
具有大数据的不均匀多孔培养基方程：
An inhomogeneous porous medium equation with large data: well-posedness
论文作者
论文摘要
我们研究欧几里得加权多孔培养基方程的溶液，当重量在[0,2）$中的$ | x |^{ - γ} $（例如$ | x |^{ - γ} $（[0,2）$时的表现时，允许在原点上奇异。特别是，我们以加权$ l^1 $ -A-A-A-A-A-avervicen sense中的意义，以$ | x |^{（2-γ）/（M-1）/（2-γ）/（M-1）/（M-1）/（M-1）/（M-1）} $以“终点”的“终点”显示出来的本地存在和独特性。我们还确定了全球存在和爆破类别，其各自的形式极大地支持了至少对于积极解决方案，这种增长率是最佳的说法。作为我们存在证明的关键步骤，我们为大数据建立了局部平滑效果，而无需诉诸于经典的Aronson-Bénilan不平等，并使用Bénilan-Crandall不平等，而是使用Bénilan-Crandall不平等，这可能具有独立的利益，因为后者在更一般的一般环境中存在。
We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spacial infinity like $|x|^{-γ}$, for $γ\in [0,2)$, and is allowed to be singular at the origin. In particular we show local-in-time existence and uniqueness for a class of large initial data which includes as "endpoints" those growing at a rate of $ |x|^{(2-γ)/(m-1)}$, in a weighted $L^1$-average sense. We also identify global-existence and blow-up classes, whose respective forms strongly support the claim that such a growth rate is optimal, at least for positive solutions. As a crucial step in our existence proof we establish a local smoothing effect for large data without resorting to the classical Aronson-Bénilan inequality and using the Bénilan-Crandall inequality instead, which may be of independent interest since the latter holds in much more general settings.
