Evolution equations involving nonlinear truncated Laplacian operators

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