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December  2010, 5(4): 745-763. doi: 10.3934/nhm.2010.5.745

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

Jérôme Coville 1, , Nicolas Dirr 2, and  Stephan Luckhaus 3,

1. 

Equipe BIOSP, INRA Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France
2. 

Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
3. 

Mathematisches Institut der Universität Leipzig, PF 100920, Leipzig, Germany

Received  October 2009 Revised  April 2010 Published  November 2010

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: Random obstacles, Qualitative behavior of parabolic PDEs with random coefficients, Interface evolution in Random media..
Mathematics Subject Classification: 35R60, 35B09, 82C4.
Citation: Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks & Heterogeneous Media, 2010, 5 (4) : 745-763. doi: 10.3934/nhm.2010.5.745
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show all references

References:
[1]

S. Brazovsii and T. Nattermann, Pinning and sliding of driven elastic systems: From domain walls to charge density waves, Advances in Physics, 53 (2004), 177-252. doi: 10.1080/00018730410001684197.  Google Scholar

[2]

L. A. Caffarelli, P. E. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361. doi: 10.1002/cpa.20069.  Google Scholar

[3]

N. Dirr, G. Karali and N. K. Yip, Pulsating wave for mean curvature flow in inhomogeneous medium, European Journal of Applied Mathematics, 19 (2008), 661-699. doi: 10.1017/S095679250800764X.  Google Scholar

[4]

N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media, Interfaces and Free Boundaries, 8 (2006), 79-109. doi: 10.4171/IFB/136.  Google Scholar

[5]

G. R. Grimmett and D. R. Stirzaker, "Probability and Random Processes," Oxford University Press, Oxford, 1992.  Google Scholar

[6]

P.-L. Lions and P. E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 667-677. doi: 10.1016/j.anihpc.2004.10.009.  Google Scholar

[7]

L. Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure Appl. Math., 6 (1953), 167-177. doi: 10.1002/cpa.3160060202.  Google Scholar

[8]

J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2.  Google Scholar

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