  We consider generalized solutions of the Perona-Malik equation in dimension
one, defined as all possible limits of solutions to the semi-discrete
approximation in which derivatives with respect to the space variable are
replaced by difference quotients.
  Our first result is a pathological example in which the initial data converge
strictly as bounded variation functions, but strict convergence is not
preserved for all positive times, and in particular many basic quantities, such
as the supremum or the total variation, do not pass to the limit. Nevertheless,
in our second result we show that all our generalized solutions satisfy some of
the properties of classical smooth solutions, namely the maximum principle and
the monotonicity of the total variation.
  The verification of the counterexample relies on a comparison result with
suitable sub/supersolutions. The monotonicity results are proved for a more
general class of evolution curves, that we call $uv$-evolutions.
