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# Existence and stability of traveling waves for an integro-differential equation for slow erosion

Abstract: We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at $\pm\infty$. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approaches the traveling waves asymptotically as $t\to+\infty$.

Paper:
Available as PDF (611 Kbytes).
Author(s):
Graziano Guerra
Wen Shen
Publishing information:
A revised version of the paper has appeared in Journal of Differential Equations 256 (2014) 253–282.
Submitted by:
; 2013-03-20.