Special Session 93: Recent trends in elliptic and parabolic equations

Very Singular Solution to nonlinear equation with absorption

Said BENACHOUR
Institut Elie Cartan - Universit\`e de Lorraine.fr
France
Co-Author(s):    Philippe Laurencot
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.