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Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation

Arturo de Pablo, Fernando Quirós, Jorge Ruiz-Cases

Abstract

We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, \[\partial_t u + \mathcal{L}\varphi(u) = 0,\] and the interplay between these two properties. Here $\mathcal{L}$ is a stable-like operator and $\varphi$ is a singular nonlinearity. We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation. The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where $\varphi$ is a power.

Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation

Abstract

We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, and the interplay between these two properties. Here is a stable-like operator and is a singular nonlinearity. We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation. The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where is a power.
Paper Structure (15 sections, 17 theorems, 124 equations, 1 figure)

This paper contains 15 sections, 17 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Goal.
  3. Main results.
  4. Outline of the proofs.
  5. Precedents.
  6. Preliminaries
  7. On the operator.
  8. On the concept of solution.
  9. On the smoothing effect.
  10. On the fundamental solution.
  11. On the regularity of solutions.
  12. Positivity of solutions
  13. From nonlinear parabolic to linear elliptic.
  14. Elliptic weak Harnack inequality.
  15. Asymptotic behaviour

Key Result

Theorem 1.1

Let $u$ be a bounded weak solution of problem Cauchy. Then, for any $p \in [1, \frac{N}{(N-\sigma)_+})$, $x_0 \in \mathbb{R}^N$, a.e. $t>0$ and $R>0$, where $c>0$ depends on $t, R$ and $\varphi$, as well as on $N$ and $\sigma$.

Figures (1)

  • Figure 1: Sketch of the proof

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 1.1: Positivity
  • Theorem 1.2: Asymptotic behaviour
  • Remark 1.2
  • Theorem 2.1: Smoothing effect
  • Remark 2.1
  • Theorem 3.1: CrandallMichaelPierre1982
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • ...and 27 more