| Abstract: |
| Abstract:\
We investigate the existence of a Very Singular Solution at the origin
to a viscous Hamilton-Jacobi equation.\
A Very Singular Solution $V$at the origin is a
non-negative solution which is smooth in
$(0,+\infty)\times R^N$ and fulfills the fact that the
singularity of $V$ in $(t,x)=(0,0)$ is stronger than the singularity
in $(t,x)=(0,0)$ of fundamental solutions, that is
the solutions whose initial data is $\delta$ ( the Dirac mass centered at $x=0$).\
Besides the description of the isolated
singularities in $(t,x)=(0,0)$ the Very Singular Solutions
(when they exist) also play an important role in the description of the
large time behaviour of the solutions of PDE.\
The name Very Singular Solution has been introduced by Brezis, Peletier and Terman in 1984. |
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