  Recently, it has been claimed (O. V. Gendelman and A. V. Savin, Phys. Rev.
Lett. {\bf 84}, 2381 (2000); A.V.Savin and O.V.Gendelman, arXiv:
cond-mat/0204631 (2002)) that two nonlinear classical 1-d lattice models show
transitions, at finite temperatures, where the heat conduction changes from
being finite to being infinite. These are the well known Frenkel-Kontorova (FK)
model and a model for coupled rotators. For the FK model we give strong
theoretical arguments why such a phase transition is not to be expected. For
both models we show numerically that the effects observed by Gendelman {\it et
al.} are not true phase transitions but are rather the expected cross-overs
associated to the conductivity divergence as $T\to 0$ and (for the FK model)
$T\to\infty$.
