  We study the interplay of neutrino oscillation and invisible decay in
atmospheric and long-baseline neutrinos experiments. We perform a global
analysis of the full atmospheric data from Super-Kamiokande together with
long-baseline K2K and MINOS in these scenarios. We find that the admixture of
nu_mu -> nu_tau oscillations with parameters Delta_m32^2 = 2.6e-3 eV^2 and
theta_23 ~ 34 degrees plus decay of the heavy neutrino, nu_3, with lifetime of
the order tau_3/m_3 ~ 2.6e-12 s/eV provides a reasonable fit to atmospheric
neutrinos, although this solution becomes more disfavored (dropping to the 99%
CL) once long-baseline data are included. Other than this local minimum, the
analysis shows no evidence in favor of a non-vanishing neutrino decay width and
an lower bound on the decay lifetime tau_3/m_3 > 9.3e-11 s/eV is set at 99% CL.
In the framework of Majoron models, this constraint can be translated into a
bound on the Majoron coupling to nu_3 and an unmixed very light sterile state,
|g_s3| < 8.6e-3 (2.2 eV/m_3).
