Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients

Pages: 745 - 763, Volume 5, Issue 4, December 2010

doi:10.3934/nhm.2010.5.745       Abstract        References        Full Text (336.1K)       Related Articles

Jérôme Coville - Equipe BIOSP, INRA Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France (email)
Nicolas Dirr - Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom (email)
Stephan Luckhaus - Mathematisches Institut der Universität Leipzig, PF 100920, Leipzig, Germany (email)

Abstract: We consider a so-called random obstacle model for the motion of a hypersurface through a field of random obstacles, driven by a constant driving field. The resulting semi-linear parabolic PDE with random coefficients does not admit a global nonnegative stationary solution, which implies that an interface that was flat originally cannot get stationary. The absence of global stationary solutions is shown by proving lower bounds on the growth of stationary solutions on large domains with Dirichlet boundary conditions. Difficulties arise because the random lower order part of the equation cannot be bounded uniformly.

Keywords:  Qualitative behavior of parabolic PDEs with random coefficients, Random obstacles, Interface evolution in Random media.
Mathematics Subject Classification:  35R60, 35B09, 82C44.

Received: October 2009;      Revised: April 2010;      Published: November 2010.

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