dissipation effects to determine the overall solution behavior. This work extends some recent contributions led by the author [@Zingano2010; @Zingano2011] (see also [@BarrionuevoOliveiraZingano2014; @BrazMeloZingano2015]) where the analysis was confined to one-dimensional problems, primarily in the case of linear advection, for which solutions are automatically global and can only misbehave by increasing unboundedly as $ t \rightarrow \infty $. Here, we consider the much more challenging case of superlinear advection (and arbitrary dimension), which may cause finite-time blow-up in several important spaces. We then point out a new kind of phenomena — one that may be properly named “anti-Fujita" for its vivid contrast to the type of blow-up behavior discovered by Fujita in the 1960s, and which has been investigated ever since — that has apparently been completely overlooked in the literature. For better clarity, we restrict ourselves here to the case of linear nondegenerate diffusion, but similar properties and behavior are also to be found in much more general diffusion phenomena, as will be reported shortly. ]{}

\
[**Índice**]{}\
[\
]{}[Parte I: Problema de Leray]{}\
\
1. Introdução $\;$1\
2. Preliminares, I $\;$7\
3. Preliminares, II $\;$12\
4. Prova de (1.12$a$) $\;$16\
5. Prova de (1.12$b$) $\;$23\
Apêndice A $\;$28\
