Barenblatt profiles for a nonlocal porous medium equation

By Piotr Biler, C. Imbert, Grzegorz Karol Karch. 2010

67 citations recorded by the source corpus.

Abstract We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d39af1ebae2d078cdbbd42299e04cb347240b9dc

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Fractional Laplace Operator and Meijer G-function (2015, background): “(36) Theorems 1 and 2 readily imply the following explicit expressions, which extend those previously derived in [2, 8] (see also [15]).”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “[21] P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math.”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “The authors of [31] found the self-similar Barenblatt profiles in all dimensions with explicit formulas: F (x) = (A−B |x|2)1−s + .”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, background): “This problem has been studied by Biler, Imbert and Karch in [1, 2].”
Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, unknown): “A formula for Δα/2u p with p ∈ (0, 1) and α ∈ (−1, 1) is announced in [6].”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, background): “Evolution problems of nonlocal diffusion type originated from material science and statistical mechanics, and led to much work on fractional porous medium models, see for instance [7, 22, 23, 20, 43, 44, 6, 10, 52, 60, 25], the work was summarized in [56, 58].”
Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, unknown): “A formula for Δα/2u p with p ∈ (0, 1) and α ∈ (−1, 1) is announced in [6].”