  We first study the so-called Heat equation with two families of elliptic
operators whichare fully nonlinear, and depend on some eigenvalues of the
Hessian matrix. The equationwith operators including the "large" eigenvalues
has strong similarities with a Heatequation in lower dimension whereas,
surprisingly, for operators including "small" eigenvalues it shares some
properties with some transport equations. In particular, forthese operators,
the Heat equation (which is nonlinear) not only does not have theproperty that
"disturbances propagate with infinite speed" but may lead to quenchingin finite
time. Last, based on our analysis of the Heat equations (for which we providea
large variety of special solutions) for these operators, we inquire on the
associated Fujita blow-up phenomena.

Analysis of PDEs