Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients
Patrick W. Dondl - Mathematical Sciences, Durham University, Science Site, South Road, Durham DH1 3LE, United Kingdom (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.
Received: March 2011; Revised: September 2011; Published: February 2012.
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