On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

DOI10.3934/dcds.2015.35.5927zbMath1336.35057arXiv1403.5293OpenAlexW1574379807MaRDI QIDQ255580

Fabio Punzo, Matteo Muratori, Gabriele Grillo

Publication date: 9 March 2016

Published in: Discrete and Continuous Dynamical Systems (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1403.5293

zbMATH Keywords

porous medium equationnonlinear diffusion equationslong time behaviourfractional LaplacianBarenblatt solutionseparable variable solutions

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Nonlinear parabolic equations (35K55) Degenerate parabolic equations (35K65) Initial value problems for second-order parabolic equations (35K15) Fractional partial differential equations (35R11) Self-similar solutions to PDEs (35C06)

Related Items (20)

Asymptotic behavior for the critical nonhomogeneous porous medium equation in low dimensions ⋮ Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation ⋮ The fractional Laplacian in power-weighted \(L^{p}\) spaces: integration-by-parts formulas and self-adjointness ⋮ Blow-up and global existence for the inhomogeneous porous medium equation with reaction ⋮ Fractional porous media equations: existence and uniqueness of weak solutions with measure data ⋮ An inhomogeneous porous medium equation with large data: well-posedness ⋮ Asymptotic decay of solutions for sublinear fractional Choquard equations ⋮ Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation. I. Stratified groups ⋮ Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights ⋮ On the Cauchy problem for a general fractional porous medium equation with variable density ⋮ The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay ⋮ Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity ⋮ The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case ⋮ One-dimensional super-fast diffusion: persistence versus extinction revisited -- extinction at spatial infinity ⋮ Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density ⋮ Some recent advances in nonlinear diffusion on negatively-curved Riemannian manifolds: from barriers to smoothing effects ⋮ Weighted fast diffusion equations II: sharp asymptotic rates of convergence in relative error by entropy methods ⋮ Long-time behavior for the porous medium equation with small initial energy ⋮ Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density ⋮ The fractional porous medium equation on manifolds with conical singularities. I

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities
• Asymptotic behavior for the heat equation in nonhomogeneous media with critical density
• On the Cauchy problem for nonlinear parabolic equations with variable density
• Hitchhiker's guide to the fractional Sobolev spaces
• A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains
• A fractional porous medium equation
• Comparison and regularity results for the fractional Laplacian via symmetrization methods
• Conditions at infinity for the inhomogeneous filtration equation
• The Cauchy problem for the inhomogeneous porous medium equation
• The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium
• Continuity of the temperature in boundary heat control problems
• Long time behavior for the inhomogeneous PME in a medium with slowly decaying density
• Asymptotics of the fast diffusion equation via entropy estimates
• Sublinear elliptic equations in \(\mathbb{R}{}^ n\)
• On the Cauchy problem for a class of parabolic equations with variable density
• Asymptotic behaviour for the porous medium equation posed in the whole space.
• Strong solutions in \(L^ 1\) of degenerate parabolic equations
• Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density
• Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
• Well-posedness for the Cauchy problem for a fractional porous medium equation with variable density in one space dimension.
• On the Cauchy problem for a general fractional porous medium equation with variable density
• Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density
• Quantitative local and global a priori estimates for fractional nonlinear diffusion equations
• Nonlinear equations for fractional Laplacians. I: Regularity, maximum principles, and Hamiltonian estimates
• Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
• Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian
• Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations
• The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions
• A concave—convex elliptic problem involving the fractional Laplacian
• A General Fractional Porous Medium Equation
• The Asymptotic Behavior of Gas in an n-Dimensional Porous Medium
• Non-linear diffusion in a finite mass medium
• Uniqueness of the solutions of ut−Δϕ(u) = 0 with initial datum a measure
• The Filtration Equation in a Class of Functions Decreasing at Infinity
• On a fractional sublinear elliptic equation with a variable coefficient
• An Extension Problem Related to the Fractional Laplacian

This page was built for publication: On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density