Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects

Abstract

We investigate forward and backward smoothing effects in Lebesgue spaces L^p and ℳ^p:=L^p,∞ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation ∂_t u+(−Δ)^σ/2|u|^m−1u=0 in ℝ^N, 0<σ<2, in the very fast range 0<m≤ m_c:=(N−σ)/N. We prove that very weak solutions have an ℳ^p--L^∞ smoothing effect if p>p^*:=N/σ(1−m), and we construct counterexamples showing the failure of any L^p--ℳ^q forward (q>p) smoothing effect if 1< p< p^*, m<m_c or L¹--ℳ^q (q>1) if m=m_c. We also prove a backward ℳ^p--L¹ smoothing effect whenever 1≤ p<p^*, m<m_c, and we construct counterexamples showing that there is no L^p--ℳ^q backward (q<p) smoothing effect if p> p^*. Regarding the threshold value p=p^*, we prove that all solutions starting in ℳ^{p^*} become extinct in finite time, and show the failure of any ℳ^{p^*}--ℳ^q smoothing before extinction for any q≠ p^*. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case σ=2.

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BibTeX RIS

BibTeX

@misc{boudaoud2026fractional,
  title = {{Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects}},
  author = {Mohammed-El-Mahdi Boudaoud and Arturo de Pablo and Fernando Quirós},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12703},
  eprint = {2609.12703},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12703},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects
AU  - Mohammed-El-Mahdi Boudaoud
AU  - Arturo de Pablo
AU  - Fernando Quirós
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12703
UR  - https://arxiv.org/abs/2609.12703
AB  - We investigate forward and backward smoothing effects in Lebesgue spaces L^p and ℳ^p:=L^p,∞ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation ∂_t u+(−Δ)^σ/2|u|^m−1u=0 in ℝ^N, 0<σ<2, in the very fast range 0<m≤ m_c:=(N−σ)/N. We prove that very weak solutions have an ℳ^p--L^∞ smoothing effect if p>p^*:=N/σ(1−m), and we construct counterexamples showing the failure of any L^p--ℳ^q forward (q>p) smoothing effect if 1< p< p^*, m<m_c or L^1--ℳ^q (q>1) if m=m_c. We also prove a backward ℳ^p--L^1 smoothing effect whenever 1≤ p<p^*, m<m_c, and we construct counterexamples showing that there is no L^p--ℳ^q backward (q<p) smoothing effect if p> p^*. Regarding the threshold value p=p^*, we prove that all solutions starting in ℳ^p^* become extinct in finite time, and show the failure of any ℳ^p^*--ℳ^q smoothing before extinction for any q≠ p^*. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case σ=2.
ER  -

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