Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights

Bonforte, Matteo; Simonov, Nikita

doi:10.4171/aihpc/42

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Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights

Bonforte, Matteo ; Simonov, Nikita

Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 1-59

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Résumé

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) $u_{t} = {| x |}^{γ} {div (| x |}^{- β} \nabla u^{m})$ posed on $(0, + \infty) \times ℝ^{d}$ , with $d \geq 3$ , in the so-called good fast diffusion range $m_{c} < m < 1$ , within the range of parameters $γ, β$ which is optimal for the validity of the so-called Caffarelli–Kohn–Nirenberg inequalities.

It is natural to ask in which sense such solutions behave like the Barenblatt $𝔅$ (fundamental solution): for instance, asymptotic convergence, i.e. ${∥ u (t) - 𝔅 (t) ∥}_{L^{p} (ℝ^{d})} \overset{t \to \infty}{\to} 0$ , is well known for all $1 \leq p \leq \infty$ , while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $𝒳 \subset L_{+}^{1} (ℝ^{d})$ that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. $d_{\infty} (u (t)) = {∥ u (t) / ℬ (t) - 1 ∥}_{L^{\infty} (ℝ^{d})} \overset{t \to \infty}{\to} 0$ . Such a characterization is in terms of an integral condition on $u (t = 0)$ .

To the best of our knowledge, analogous issues for the linear heat equation, $m = 1$ , do not possess such clear answers, but only partial results. Our characterization is also new for the classical, nonweighted FDE. We are able to provide minimal rates of convergence to $ℬ$ in different norms. Such rates are almost optimal in the nonweighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in $L_{+}^{1} (ℝ^{d}) ∖ 𝒳$ , preserve the same “fat” spatial tail for all times, hence UREC fails and $d_{\infty} (u (t)) = \infty$ , even if ${∥ u (t) - ℬ (t) ∥}_{L^{1} (ℝ^{d})} \overset{t \to \infty}{\to} 0$ .

Accepté le : 2021-11-25
Publié le : 2022-07-27

DOI : 10.4171/aihpc/42

Classification : 35K55, 35B40, 35B45, 35K67, 35K65
Mots-clés : Caffarelli–Kohn–Nirenberg weights, global Harnack inequalities, tail behaviour, Fast diffusion equation, asymptotic behaviour

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     title = {Fine properties of solutions to the {Cauchy} problem for a fast diffusion equation with {Caffarelli{\textendash}Kohn{\textendash}Nirenberg} weights},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     doi = {10.4171/aihpc/42},
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Bonforte, Matteo; Simonov, Nikita. Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 1-59. doi: 10.4171/aihpc/42

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