Abstract
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.