Evolution equations involving nonlinear truncated Laplacian operators

Matthieu Alfaro (1) , Isabeau Birindelli (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

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.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems - Series A, 2020, 40 (6), pp.3057-3073. ⟨10.3934/dcds.2020046⟩. ⟨hal-02077996⟩