  We study a dissipative nonlinear equation modelling certain features of the
Navier-Stokes equations. We prove that the evolution of radially symmetric
compactly supported initial data does not lead to singularities in dimensions
$n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting
existence of blow-up solutions. Moreover, using the same techniques we
numerically confirm a conjecture of Lepin regarding existence of self-similar
singular solutions to a semi-linear heat equation.
