Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights
We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) ut = |x| div(|x|-β ∇um) posed on (0, +1) × ℝd, with d ≥ 3, in the so-called good fast diffusion range mc < m < 1, within the range of parameters γ, β w...
| Autores: | ; |
|---|---|
| Tipo de documento: | Artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Lenguaje: | Inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/707989 |
| Acceso en línea: | http://hdl.handle.net/10486/707989 https://dx.doi.org/10.4171/AIHPC/42 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| _version_ | 1878738251366268928 |
|---|---|
| author | Bonforte, Matteo |
| author2 | Simonov, Nikita |
| author2_role | author |
| author_browse | Bonforte, Matteo Simonov, Nikita |
| author_facet | Bonforte, Matteo Simonov, Nikita |
| author_role | author |
| collection | Biblos-e Archivo. Repositorio Institucional de la UAM |
| description | We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) ut = |x| div(|x|-β ∇um) posed on (0, +1) × ℝd, with d ≥ 3, in the so-called good fast diffusion range mc < 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)∥Lp(ℝd)t→∞ 0, is well known for all 1 ≤ p ≤ 1, while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X ⊂ L1+(ℝd) that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. d∞(u(t)) = ∥u(t)=B(t) - 1∥L∞(ℝd)t→∞ 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 L1+(ℝd) n X, preserve the same “fat” spatial tail for all times, hence UREC fails and d∞(u(t)) = 1, even if ∥u(t) - B(t)∥L1(ℝd)t→∞ 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_abf73620fec2455013c58b6cd4c7ef65 |
| institution | Universidad Autónoma de Madrid |
| instname_str | Universidad Autónoma de Madrid |
| language | eng |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:repositorio.uam.es:10486/707989 |
| publishDate | 2022 |
| publishDateSort | 2022 |
| publisher | European Mathematical Society Publishing House |
| publisherStr | European Mathematical Society Publishing House |
| reponame_str | Biblos-e Archivo. Repositorio Institucional de la UAM |
| spelling | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights Bonforte, Matteo Simonov, Nikita asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) ut = |x| div(|x|-β ∇um) posed on (0, +1) × ℝd, with d ≥ 3, in the so-called good fast diffusion range mc < 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)∥Lp(ℝd)t→∞ 0, is well known for all 1 ≤ p ≤ 1, while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X ⊂ L1+(ℝd) that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. d∞(u(t)) = ∥u(t)=B(t) - 1∥L∞(ℝd)t→∞ 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 L1+(ℝd) n X, preserve the same “fat” spatial tail for all times, hence UREC fails and d∞(u(t)) = 1, even if ∥u(t) - B(t)∥L1(ℝd)t→∞ 0 This work was partially funded by Projects MTM2017-85757-P and PID2020- 113596GB-I00 (Spain) and by the EU H2020 MSCA programme, grant agreement 777822. M.B. acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S). N.S. was partially funded by the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Spain) and also by Regione Ile-de-France. This work has been partially supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency European Mathematical Society Publishing House http://hdl.handle.net/10486/707989 https://dx.doi.org/10.4171/AIHPC/42 |
| spellingShingle | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights Bonforte, Matteo asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| title | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title_full | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title_fullStr | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title_full_unstemmed | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title_short | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title_sort | Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| topic | asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| topic_facet | asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| url | http://hdl.handle.net/10486/707989 https://dx.doi.org/10.4171/AIHPC/42 |