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

Coville, Jérôme, Dirr, Nicolas P. and Luckhaus, Stephan 2010. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks and Heterogeneous Media 5 (4) , pp. 745-763. 10.3934/nhm.2010.5.745

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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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Qualitative behavior of parabolic PDEs with random coefficients, Random obstacles, Interface evolution in Random media
Publisher:	American Institute of Mathematical Sciences
ISSN:	1556-1801
Last Modified:	04 Jun 2017 02:52
URI:	http://orca.cf.ac.uk/id/eprint/13072

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