A Geometrical Approach to the Study of Unbounded
Solutions of Quasilinear Parabolic Equations

Guy Barles, Samuel Biton et Olivier Ley

In this article, we are interested in the existence and
uniqueness of solutions for quasilinear parabolic equations set in the
whole space RN. We consider in particular cases when
there is no restriction on the growth or the behaviour of these
solutions at infinity. Our model equation is the mean
curvature equation for graphs for which Ecker and Huisken have
shown the existence of smooth solutions for any locally
Lipschitz continuous initial data. We use a geometrical
approach which consists in seeing the evolution of the
graph of a solution as a geometric motion which is then studied by
the so-called "level-set approach." After determining
the right class of quasilinear parabolic pdes which can be taken into
account by this approach, we show how the uniqueness
for the original pde is related to "fattening phenomena" in the
level-set approach. Existence of solutions is proved
using a local Linfinity-bound obtained by using in an essential way
the level-set approach. Finally we apply these results
to convex initial datas and prove existence and comparison results in
full generality, i.e. without restriction on their growth
at infinity.