  For evolutive Hamilton-Jacobi equations, we propose a refined definition of
C^0-variational solution, adapted to Cauchy problems for continuous initial
data. In this weaker framework we investigate the Markovian (or semigroup)
property for these solutions. In the case of p-convex Hamiltonians, when
variational solutions are known to be identical to viscosity solutions, we
verify directly the Markovian property by using minmax techniques. In the
non-convex case, we construct an explicit evolutive example where minmax and
viscous solutions are different. Provided the initial data allow for the
separation of variables, we also detect the Markovian property for
convex-concave Hamiltonians. In this case, and for general initial data, we
finally give upper and lower Hopf-type estimates for the variational solutions.
