  Motivated by the question of existence of global solutions, we obtain
pointwise upper bounds for radially symmetric and monotone solutions to the
homogeneous Landau equation with Coulomb potential. The estimates say that blow
up in the $L^\infty$-norm at a finite time $T$ can occur only if the
$L^{3/2}$-norm of the solution concentrates for times close to $T$. The bounds
are obtained using the comparison principle for the Landau equation and for the
associated mass function.
  This method provides long-time existence results for the isotropic version of
the Landau equation with Coulomb potential, recently introduced by Krieger and
Strain.
