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: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/684842 |
| Acceso en línea: | http://hdl.handle.net/10486/684842 https://dx.doi.org/10.2140/apde.2018.11.945 |
| Access Level: | acceso abierto |
| Palabra clave: | A priori estimates Boundary behavior Bounded domains Harnack inequalities Nonlinear equations Positivity Regularity Nonlocal diffusion Matemáticas |
| Sumario: | 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. |
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