A new approach to initial traces in nonlinear filtration

Abstract

For weak solutions of equations of the type of nonlinear filtration in ${\mathbf{R}}^N\times (0,T)$, $0<T<\infty$, we prove precise sup-estimates and local and global Harnack type inequalities. These estimations permit to identify the initial traces and describe the behavior of such solutions as $|x|\to \infty$.

The main point is to introduce a new approach, free of the specific features of the porous medium equation such as homogeneity, scaling, quasi-convexity, etc. This approach on one hand allows generalizations to a large variety of equations and on other yields new results on gradient averages.

Résumé

On démontre des inégalités de Harnack locales et globales, ainsi des majorations pour des solutions faibles des équations de filtrage non linéaire dans ${\mathbf{R}}^N\times (0,T)$, pour $T<\infty$. Ces estimations permettent d’identifier les traces initiales et de décrire le comportement des solutions quand $|x|\to \infty$.