On a Differential Model for Growing Sandpiles with Non-Regular Sources

Abstract

We consider a variational model that describes the growth of a sandpile on a bounded table under the action of a vertical source. The possible equilibria of such a model solves a boundary value problem for a system of nonlinear partial differential equations that we analyze when the source term is merely integrable. In addition, we study the asymptotic behavior of the dynamical problem showing that the solution converges asymptotically to an equilibrium that we characterize explicitly.

Keywords:

• Asymptotic profile
• Distance function
• Granular matter
• Uniqueness of solutions

Mathematics Subject Classification:

• Primary 35C15, 35F30
• Secondary 47J20, 35Q99

Acknowledgments

We thank Graziano Crasta for stimulating discussions and for bringing to our attention the behavior described in Remark 4.9. We also thank the anonymous referees for their useful comments. This work has been partially supported by the Italian PRIN 2005 and PRIN 2007 Programs “Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”.