论文标题
在完整的Riemannian歧管上涉及非本地算子及其在分数多孔培养基方程中的功能不平等现象
Functional Inequalities involving Nonlocal Operators on Complete Riemannian Manifolds and Their Applications to The Fractional Porous Medium Equation
论文作者
论文摘要
本文的目的是双重的。首先，我们仔细研究了涉及分数拉普拉斯运算符的各种功能性不平等，包括非本地Sobolev-Poincaré，Nash，SuperPoincaré和Goolgarithmic Sobolev型不平等，以及完全满足一些轻度的几何学假设。其次，基于派生的非局部功能不平等，我们分析了对分数多孔培养基方程的渐近行为，$ \ partial_t u +（ - Δ）此外，我们在任意完整的Riemannian歧管上建立了方程式的全球辅助性。
The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, $\partial_t u +(-Δ)^σ(|u|^{m-1}u )=0 $ with $m>0$ and $σ\in (0,1)$. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.
