  We establish that a viscosity solution to a multidimensional Hamilton-Jacobi
equation with Bohr almost periodic initial data remains to be spatially almost
periodic and the additive subgroup generated by its spectrum does not increase
in time. In the case of one space variable and a non-degenerate hamiltonian we
prove the decay property of almost periodic viscosity solutions when time
$t\to+\infty$. For periodic solutions the more general result is proved on
unconditional asymptotic convergence of a viscosity solution to a traveling
wave.
