Evolution equations involving nonlinear truncated Laplacian operators

Matthieu Alfaro; Isabeau Birindelli

doi:10.3934/dcds.2020046

Article Contents

2020, Volume 40, Issue 6: 3057-3073. Doi: 10.3934/dcds.2020046 open access

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Evolution equations involving nonlinear truncated Laplacian operators

Matthieu Alfaro^1, , and
Isabeau Birindelli^2,

1.
IMAG, Univ. Montpellier, CNRS, Montpellier, France
2.
Dipartimento di Matematica Guido Castelnuovo, Sapienza Universitaà di Roma, 00185, Roma, Italia

^* Corresponding author
^* Corresponding author

Received: February 28, 2019

Published: March 2020

Matthieu Alfaro is supported by the ANR I-SITE MUSE, project MICHEL 170544IA (n° ANR IDEX-0006) I. Birindelli is supported by INDAM-Gnampa and Ateneo Sapienza

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K05, 35K65; Secondary: 35L02, 35C06, 35D40.

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