Pinning and De-Pinning Phenomena in Front Propagation in Heterogeneous Media

revised version: November 2005
Nicolas Dirr, and Aaron Nung Kwan Yip
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Submission date: 13. May. 2005
Pages: 34
published in: Interfaces and free boundaries, 8 (2006) 1, p. 79-109 
MSC-Numbers: 35B27, 35B40, 35K55, 35K57
Keywords and phrases: reaction-diffusion, pinning, propagation failure, pulsating waves
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Abstract:
This paper investigates the pinning and de-pinning phenomena of some evolutionary partial differential equations which arise in the modeling of the propagation of phase boundaries in materials under the combined effects of an external driving force F and an underlying heterogeneous environment. The phenomenology is the existence of pinning states -- stationary solutions -- for small values of F, and the appearance of genuine motion when F is above some threshold value. In the case of a periodic medium, we characterize quantitatively, near the transition regime, the scaling behavior of the interface velocity as a function of F. The results are proved for a class of some semi-linear and reaction-diffusion equations.