Asymptotic behaviour for the porous medium equation posed in the whole space

By Juan Luis V'azquez. 2003

185 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b8f308b314c0a09a02aec9f85a16e3ee397bef63

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, unknown): “On asymptotic behaviour: [240] and [241].”
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.”
Asymptotics of the porous media equation via Sobolev inequalities (2005, background): “[8], [20], [24], [29], [32], [33], [34], [35], whose arguments are valid in the present context as well.”
Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds (2008, unknown): “[32,34].”
Global positivity estimates and Harnack inequalities for the fast diffusion equation (2006, unknown): “In the limit of the upper and lower estimates for large times we arrive at the asymptotic convergence in relative error introduced in [25].”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, methodology): “This is the main asymptotic theorem for the PME, proved in complete form by Vazquez in 2001, [65], expanding on the result by Friedman and Kamin, 1980, [40], where the authors took u0 ≥ 0, with compact support, and f = 0.”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, background): “ t 1 >, ">e 0, and %e 0 >0, such that u(x;t 1) v(x;t 1) eu 0(x); for all x2RN; which is our thesis. We ask the reader to note that improved global positivity estimates were proved in [36, 52] and [11] for the Porous Medium Equation. Now, with the next crucial lemma, we prove that the e”
Finite speed of propagation in porous media by mass transportation methods (2004, background): “…the problems Pn ut = (u m)xx, x ∈ R, t > 0, m > 1, (4) u(x, 0) = u0n(x), x ∈ R, (5) where u0n(x), n ∈ N, is a sequence of bounded, integrable and strictly positive C ∞- smooth functions such that all their derivatives are bounded in R, the condition (m − 1)(u0 m n )xx ≥ −au0n holds for some…”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, unknown): “also [7], [9], [12], [14], [16], [20], [22], [24], [26].”
Duality and Stability for Functional Inequalities (2016, background): “Within this range, the flow associated to the fast diffusion equation is in many ways even better than the flow associated to the heat equation; see Vázquez [20].”