  We consider the Cauchy problem for a multidimensional scalar conservation law
and construct an outer estimate for the domain of dependence of its Kruzkov
solution. The estimate can be represented as the controllability set of a
specific differential inclusion. In addition, reachable sets of this inclusion
provide outer estimates for the support of the wave profiles. Both results
follow from a modified version of the classical Kruzkov uniqueness theorem,
which we also present in the paper. Finally, the results are applied to a
control problem consisting in steering a distributed quantity to a given set.
