Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density

Iagar, Razvan Gabriel; Sánchez, Ariel

doi:10.1016/j.na.2014.02.016

Cited by 17 publications

(21 citation statements)

References 22 publications

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“…In the standard examples the weight is a logarithmic correction of the critical density ρ(x) = |x| −2 , a case where the mathematical analysis is particularly difficult due to the complete homogeneity under scaling, see e.g. [20,17].…”

Section: +µmentioning

confidence: 99%

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

Grillo

Muratori

Vázquez

2017

Advances in Mathematics

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We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows.We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

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Section: +µmentioning

confidence: 99%

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

Grillo

Muratori

Vázquez

2017

Advances in Mathematics

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“…This strategy has been used in [10] to deal with the PME on the class of negatively curved manifolds discussed here. It turns out that the kind of supersolution used here can be guessed from the known asymptotics of the corresponding weighted, Euclidean heat equation with critical weight as considered in [15] (see [16] for a generalization to the corresponding weighted PME). Such a supersolution is strictly related to the bound from below on the Ricci curvature, via the growth of the measure of the sphere as the radius increases.…”

Section: Introductionmentioning

confidence: 99%

The porous medium equation with large initial data on negatively curved Riemannian manifolds

Grillo

Muratori

Punzo

2018

Journal de Mathématiques Pures et Appliquées

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Abstract. We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at infinity too fast. This proves in particular that the growth rate of the data we consider is optimal for existence. Pointwise blow-up is also shown for a particular class of manifolds and of initial data.

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“…Under the same assumptions as in Theorem 3.1, with in addition d > 4s and γ > 4s, thanks to the uniqueness results of Theorem 2.3 and Theorem 4.4, we can read the above asymptotic result as follows: any nontrivial local strong solution u to (1) satisfying u m ∈ L 1 (1+|x|) −d+2s (R d × (t 0 , T )) for all T > t 0 > 0 converges, in the sense of (25), to the unique nontrivial local weak and very weak solution w to (10) …”

Section: Remarkmentioning

confidence: 94%

“…In fact, in such case, we are not able either to construct the asymptotic profile as in (i) or the minimal solution to the sublinear elliptic equation as in (ii). Observe that for s = 1 the long time behaviour of solutions has been investigated in [24] for m = 1, and in [29,25] for m > 1. For 0 < s < 1 and γ = 2s, the asymptotic behaviour of solutions is then an interesting open problem.…”

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confidence: 99%

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

Grillo¹,

Muratori²,

Punzo³

2015

DCDS-A

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We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation. 5927 5928 GABRIELE GRILLO, MATTEO MURATORI AND FABIO PUNZO ASYMPTOTIC BEHAVIOUR FOR THE FRACTIONAL PME 5929 and lim t→∞ t α u(t) − u * M (t) ∞ = 0 .Here, u * M is the self-similar Barenblatt solution of mass

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Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density

Cited by 17 publications

References 22 publications

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

The porous medium equation with large initial data on negatively curved Riemannian manifolds

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

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