The Cauchy problem for a system of Maxwell equations.
The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation
We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.
Three cylinder inequalities and unique continuation properties for parabolic equations
We prove the following unique continuation property. Let be a solution of a second order linear parabolic equation and a segment parallel to the -axis. If has a zero of order faster than any non constant and time independent polynomial at each point of then vanishes in each point, , such that the plane has a non empty intersection with .
Transformation de Kelvin et ondes non linéaires globales