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The fractional porous medium equation on the hyperbolic space
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-09-15 , DOI: 10.1007/s00526-020-01817-2
Elvise Berchio , Matteo Bonforte , Debdip Ganguly , Gabriele Grillo

We consider a 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 \(\Delta _{\mathbb {H}^{N}}\), 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.



中文翻译:

双曲空间上的分数多孔介质方程

我们考虑一个多孔介质类型的非线性简并抛物线方程,其扩散是由双曲空间上的(谱)分数拉普拉斯算子驱动的。我们在适当的弱意义上为属于普通\(L ^ p \)空间或较大的(加权)空间的数据(根据基态\(\ Delta _ { \ mathbb {H} ^ {N}} \)或(分数)Green函数的值。对于此类解决方案,我们还以定量\(L ^ 1-L ^ \ infty \)估计的形式证明了不同类型的平滑效果。据我们所知,这似乎是首次在非紧缩,几何非平凡实例上处理了分数多孔介质方程。

更新日期:2020-09-16
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