# American Institute of Mathematical Sciences

September  2015, 35(9): 4225-4239. doi: 10.3934/dcds.2015.35.4225

## Global propagation of singularities for time dependent Hamilton-Jacobi equations

 1 Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma 2 CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France 3 Dipartimento di Matematica, Università di Roma, Via della Ricerca Scientifica 1, 00133 Roma

Received  August 2014 Revised  October 2014 Published  April 2015

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form $$\label{abstract:EQ} u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in (0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,.$$ It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian $H$. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of (1) cannot vanish in a finite time.
Citation: Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225
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