# Evolution equations involving nonlinear truncated Laplacian operators

## TL;DR

This paper investigates nonlinear heat equations involving truncated Laplacian operators, revealing diverse behaviors such as finite-time quenching and connections to transport equations, and explores associated blow-up phenomena.

## Contribution

It introduces and analyzes nonlinear heat equations with eigenvalue-dependent elliptic operators, uncovering novel properties and solution behaviors, including finite-time quenching and blow-up phenomena.

## Key findings

- Operators with large eigenvalues resemble lower-dimensional heat equations.
- Operators with small eigenvalues exhibit properties similar to transport equations.
- The study identifies conditions leading to finite-time quenching and blow-up phenomena.

## 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.