Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains
We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous mediumtype of the form δtu +Lum= 0, m > 1, where the operator L belongs to a general class of linear operators, and the equation is posed in a bounded domain Ω⊂RN. As possible operators we incl...
| Autores: | , , |
|---|---|
| Formato: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/684842 |
| Acesso em linha: | http://hdl.handle.net/10486/684842 https://dx.doi.org/10.2140/apde.2018.11.945 |
| Access Level: | acceso abierto |
| Palavra-chave: | A priori estimates Boundary behavior Bounded domains Harnack inequalities Nonlinear equations Positivity Regularity Nonlocal diffusion Matemáticas |
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Sharp global estimates for local and nonlocal porous medium-type equations in bounded domainsBonforte, MatteoFigalli, AlessioVázquez Suárez, Juan LuisA priori estimatesBoundary behaviorBounded domainsHarnack inequalitiesNonlinear equationsPositivityRegularityNonlocal diffusionMatemáticasWe provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous mediumtype of the form δtu +Lum= 0, m > 1, where the operator L belongs to a general class of linear operators, and the equation is posed in a bounded domain Ω⊂RN. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, L can be a fractional power of a uniformly elliptic operator with C1coefficients. Since the nonlinearity is given by umwith m > 1, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations was recently developed by Bonforte and Vázquez (2015, 2016). Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when L is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of nonmatching powers for the long-time boundary behavior. More precisely, when L =(-Δ)sis a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that • when 2s > 1-1/m, for large times all solutions behave as dist1/mnear the boundary; • when 2s ≤ 1-1/m, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation Lum= u.M.B. and J.L.V. are partially funded by Project MTM2011-24696 and MTM2014-52240-P(Spain). A.F. has been supported by NSF Grants DMS-1262411 and DMS-1361122, and by the ERC Grant \Regularity and Stability in Partial Diferential Equations (RSPDE)"MSP: Mathematical Sciences PublishersDepartamento de MatemáticasFacultad de Ciencias20182018-01-12research articlehttp://purl.org/coar/resource_type/c_2df8fbb1AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/684842https://dx.doi.org/10.2140/apde.2018.11.945reponame: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:10486/6848422026-06-23T12:46:27Z |
| dc.title.none.fl_str_mv |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| title |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| spellingShingle |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains Bonforte, Matteo A priori estimates Boundary behavior Bounded domains Harnack inequalities Nonlinear equations Positivity Regularity Nonlocal diffusion Matemáticas |
| title_short |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| title_full |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| title_fullStr |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| title_full_unstemmed |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| title_sort |
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains |
| dc.creator.none.fl_str_mv |
Bonforte, Matteo Figalli, Alessio Vázquez Suárez, Juan Luis |
| author |
Bonforte, Matteo |
| author_facet |
Bonforte, Matteo Figalli, Alessio Vázquez Suárez, Juan Luis |
| author_role |
author |
| author2 |
Figalli, Alessio Vázquez Suárez, Juan Luis |
| author2_role |
author author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemáticas Facultad de Ciencias |
| dc.subject.none.fl_str_mv |
A priori estimates Boundary behavior Bounded domains Harnack inequalities Nonlinear equations Positivity Regularity Nonlocal diffusion Matemáticas |
| topic |
A priori estimates Boundary behavior Bounded domains Harnack inequalities Nonlinear equations Positivity Regularity Nonlocal diffusion Matemáticas |
| description |
We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous mediumtype of the form δtu +Lum= 0, m > 1, where the operator L belongs to a general class of linear operators, and the equation is posed in a bounded domain Ω⊂RN. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, L can be a fractional power of a uniformly elliptic operator with C1coefficients. Since the nonlinearity is given by umwith m > 1, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations was recently developed by Bonforte and Vázquez (2015, 2016). Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when L is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of nonmatching powers for the long-time boundary behavior. More precisely, when L =(-Δ)sis a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that • when 2s > 1-1/m, for large times all solutions behave as dist1/mnear the boundary; • when 2s ≤ 1-1/m, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation Lum= u. |
| publishDate |
2018 |
| dc.date.none.fl_str_mv |
2018 2018-01-12 |
| dc.type.none.fl_str_mv |
research article http://purl.org/coar/resource_type/c_2df8fbb1 AM http://purl.org/coar/version/c_ab4af688f83e57aa |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/10486/684842 https://dx.doi.org/10.2140/apde.2018.11.945 |
| url |
http://hdl.handle.net/10486/684842 https://dx.doi.org/10.2140/apde.2018.11.945 |
| 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 |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 |
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openAccess |
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application/pdf |
| dc.publisher.none.fl_str_mv |
MSP: Mathematical Sciences Publishers |
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MSP: Mathematical Sciences Publishers |
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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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