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...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | 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 |
| Access Level: | acceso abierto |
| Palabra clave: | asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
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Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weightsBonforte, MatteoSimonov, Nikitaasymptotic behaviourCaffarelli–Kohn–Nirenberg weightsFast diffusion equationglobal Harnack inequalitiestail behaviourMatemáticasWe 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→∞ 0This 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 AgencyEuropean Mathematical Society Publishing HouseDepartamento de MatemáticasFacultad de Ciencias20222022-07-27research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/707989https://dx.doi.org/10.4171/AIHPC/42reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/7079892026-06-23T12:46:27Z |
| dc.title.none.fl_str_mv |
Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| title |
Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| 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_short |
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_sort |
Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights |
| dc.creator.none.fl_str_mv |
Bonforte, Matteo Simonov, Nikita |
| author |
Bonforte, Matteo |
| author_facet |
Bonforte, Matteo Simonov, Nikita |
| author_role |
author |
| author2 |
Simonov, Nikita |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemáticas Facultad de Ciencias |
| dc.subject.none.fl_str_mv |
asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| topic |
asymptotic behaviour Caffarelli–Kohn–Nirenberg weights Fast diffusion equation global Harnack inequalities tail behaviour Matemáticas |
| 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 |
| publishDate |
2022 |
| dc.date.none.fl_str_mv |
2022 2022-07-27 |
| dc.type.none.fl_str_mv |
research article http://purl.org/coar/resource_type/c_2df8fbb1 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/10486/707989 https://dx.doi.org/10.4171/AIHPC/42 |
| url |
http://hdl.handle.net/10486/707989 https://dx.doi.org/10.4171/AIHPC/42 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 |
| dc.rights.openaire.fl_str_mv |
info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
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application/pdf |
| dc.publisher.none.fl_str_mv |
European Mathematical Society Publishing House |
| publisher.none.fl_str_mv |
European Mathematical Society Publishing House |
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reponame:Biblos-e Archivo. Repositorio Institucional de la UAM instname:Universidad Autónoma de Madrid |
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Universidad Autónoma de Madrid |
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Biblos-e Archivo. Repositorio Institucional de la UAM |
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Biblos-e Archivo. Repositorio Institucional de la UAM |
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