Global positivity estimates and Harnack inequalities for the fast diffusion equation

By Matteo Bonforte, Juan Luis V'azquez. 2006

76 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a49bbd70e06132168611a31d5f4518d81ef453c0

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Citations in the corpus

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations (2008, methodology): “An extension work has been done in [8] for mc < m < 1 but the method collapses for m mc due to the very different properties of the solutions.”
Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation (2009, methodology): “The proof is a straightforward adaptation of the proof of the corresponding local Aleksandrov principle for the fast diffusion equation, given by two of the authors in [11].”
Quantitative local and global a priori estimates for fractional nonlinear diffusion equations (2012, methodology): “In the limit s→ 1 we recover the well known L1 local estimates for the standard FDE.”
The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour (2016, methodology): “This is a global Harnack principle, using a terminology introduced, in the case of Euclidean fast diffusion, in [7], [4].”
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation (2011, background): “Then, regularity estimates from [8] permit one to integrate by parts and prove that limt→∞F [ρ(·, t)] = C(M), which leads to (3.”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, methodology): “our works [6, 9, 10, 11]; the above estimates are surprisingly valid for m > 1 and to our knowledge they have never been observed before.”
The Fisher–KPP problem with doubly nonlinear “fast” diffusion (2016, background): “We ask the reader to note that improved global positivity estimates were proved in [36, 52] and [11] for the Porous Medium Equation.”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, result): “and L 1 data u0, with even better convergence result in relative error, refer to [ 8 ,36].”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “Symmetrization techniques were first applied to PDEs involving fractional Laplacian operators in the paper [24], where the linear elliptic case is studied.”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “Obtaining valid versions of the Harnack inequality was challenging in this range [48, 49], see also the monograph [110].”
Asymptotics of the Fast Diffusion Equation via Entropy Estimates (2007, background): “As in [10,11,47], define w(t, x) := v(t, x) VD∗(x) ∀(t, x) ∈ (0,∞)× Rd .”
Fast diffusion flow on manifolds of nonpositive curvature (2008, methodology): “We shall use them to provide lower bounds on the FET, as first noticed in [6].”