论文标题
双曲线空间上的分数多孔培养基方程
The Fractional Porous Medium Equation on the hyperbolic space
论文作者
论文摘要
我们考虑多孔培养基类型的非线性退化抛物线方程，其扩散是由双曲线空间上的（光谱）分数拉普拉斯驱动的。我们为属于通常的$ l^p $空间的数据或较大的（加权）空间的数据提供了解决方案的存在结果。对于这种解决方案，我们还以定量$ l^1-l^\ infty $估计的形式证明了不同的平滑效果。据我们所知，这似乎是第一次在非紧缩的，几何的非平凡的例子上处理分数多孔培养基方程。
We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $Δ_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.
