Calculus of Variations and Geometric Measure Theory

L. Giacomelli

Finite speed of propagation and waiting-time phenomena for degenerate parabolic equations with linear growth Lagrangian

created by giacomelli on 12 Nov 2013
modified on 05 Jul 2016


Published Paper

Inserted: 12 nov 2013
Last Updated: 5 jul 2016

Journal: SIAM J. Math. Anal.
Year: 2015


We consider a class of degenerate parabolic equations with linear growth Lagrangian. Two prototypes within this class, sharing common features with nonlinear transport equations, are the relativistic porous medium equation and the speed-limited (or flux-limited) porous medium equation. In arbitrary space dimension we prove that entropy solutions to the Cauchy problem satisfy the finite speed of propagation property, with upper bounds that we expect to be sharp. For the two aforementioned prototypes, in one space dimension we provide a condition on the growth of the initial datum which guarantees the occurrence of a waiting-time phenomenon; we also present a heuristic argument in favor of the optimality of such condition.

Keywords: entropy solutions, Degenerate parabolic equations, Singular parabolic equations, Nonlinear transport equations, Finite speed of propagation, Waiting time phenomena