Data: 28/09/2016 (quarta feira)
Hora: 13:30h
Local: C-119

Palestrante: Flávio Dickstein (IM-UFRJ)

Título: Má-colocação da equação do calor não-linear

Resumo: 
Dado α > 0 a equa¸c˜ao do calor n˜ao-linear ´e dada por ∂tu − ∆u = |u| αu. E bem sabido que o problema de Cauchy ´e bem-posto em ´ L p (R N ) se p > Nα 2 . Se α > 2 N e p < Nα 2 o problema ´e mal-posto. Discutiremos esta quest˜ao. Mostraremos, em particular, a existˆencia de infinitas solu¸c˜oes n˜ao-positivas que partem de um mesmo dado inicial positivo. Este resultado foi obtido em colabora¸c˜ao com Thierry Cazenave, Ivan Naumkin, Fred Weissler e contou com a inestim´avel ajuda de Bernardo Freitas Costa.

Data: 21/09/2016 (quarta feira)
Hora: 13:30 h
Local: C-119

Palestrante: Cristian Cazacu (Politehnica University of Bucharest)

Título: Asymptotic properties for a nonlocal diffusion problem with subcritical local convective term

Resumo: In this talk we discuss the large time behavior of the entropy
solution for an evolution model with a nonlocal diffusion part and a classical
subcritical convective term. In spite of the nonlocal diffusion, we obtain an
Oleinik type estimate similar to the case when the diffusion is local. Using
scaling arguments and hyperbolic estimates given by Oleinik's inequality, we
obtain the first term in the asymptotic behavior of the nonnegative solutions.
Finally, the large time behavior of changing sign solutions is proved using
the classical flux-entropy method and estimates for the nonlocal operator.
More precisely, we show that, as time becomes large, the nonlocal diffusion
term can be neglected and, in contrast to the supercritical case, the
asymptotic behavior is given by the convective part. Actually, this phenomenon
also occurs for the same problem with local diffusion.

This talk is based on a joint work with Liviu Ignat (Institute of Mathematics
of the Romanian Academy, Romania) and Ademir Pazoto (Universidade Federal Rio de Janeiro, Brazil).

Data: 14/09/2016 (quarta feira)
Hora: 13:30 h
Local: C-119

Palestrante: Emanuel Carneiro (IMPA)

Título: Regularity of fractional maximal operators

Resumo: In this talk we discuss the recent developments on the regularity
theory of maximal operators, when these act on Sobolev and BV functions. Among
other things, we shall discuss boundedness and continuity results for the
one-dimensional fractional Hardy-Littlewood maximal operator in the continuous
and discrete settings.
This is based on a joint work with Jose Madrid (Aalto University - Finland).