  We consider a $2\times 2$ system of hyperbolic balance laws, in one-space
dimension, that describes the evolution of a granular material with slow
erosion and deposition. The dynamics is expressed in terms of the thickness of
a moving layer on top and of a standing layer at the bottom. The system is
linearly degenerate along two straight lines in the phase plane and genuinely
nonlinear in the subdomains confined by such lines. In particular, the
characteristic speed of the first characteristic family is strictly increasing
in the region above the line of linear degeneracy and strictly decreasing in
the region below such a line. The non dissipative source term is the product of
two quantities that are transported with the two different characteristic
speeds.
  The global existence of entropy weak solutions of the Cauchy problem for such
a system was established by Amadori and Shen for initial data with bounded but
possibly large total variation, under the assumption that the initial height of
the moving layer be sufficiently small.
  In this paper we establish the Lipschitz ${\bf L^1}$-continuous dependence of
the solutions on the initial data with a Lipschitz constant that grows
exponentially in time. The proof of the ${\bf L^1}$-stability of solutions is
based on the construction of a Lyapunov like functional equivalent to the ${\bf
L^1}$-distance, in the same spirit of the functional introduced by Liu and Yang
and then developed by Bressan, Liu, Yang for systems of conservation laws with
genuinely nonlinear or linearly degenerate characteristic fields.
