Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

Pages: 137 - 150, Volume 7, Issue 1, March 2012

doi:10.3934/nhm.2012.7.137       Abstract        References        Full Text (382.7K)       Related Articles

Patrick W. Dondl - Mathematical Sciences, Durham University, Science Site, South Road, Durham DH1 3LE, United Kingdom (email)
Michael Scheutzow - Fakultät II, Institut für Mathematik, Sekr. MA 7–5, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany (email)

Abstract: We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.

Keywords:  Interfaces, heterogeneous media, random media, semilinear parabolic equations with randomness, asymptotic behavior of non-negative solutions.
Mathematics Subject Classification:  Primary: 35K58; Secondary: 35R60, 35B40, 60H15.

Received: March 2011;      Revised: September 2011;      Published: February 2012.

 References