  Despite evidence for the existence of dark matter (DM) from very high and low
redshifts, a moderate amount of DM particle decay remains a valid possibility.
This includes both models with very long-lived yet unstable particles or mixed
scenarios where only a small fraction of dark matter is allowed to decay. In
this paper, we investigate how DM particles decaying into radiation affect
non-linear structure formation. We look at the power spectrum and its redshift
evolution, varying both the decay lifetime ($\tau$) and the fraction of
decaying-to-total dark matter ($f$), and we propose a fitting function that
reaches sub-percent precision below $k\sim10$ h/Mpc. Based on this fit, we
perform a forecast analysis for a Euclid-like weak lensing (WL) survey,
including both massive neutrino and baryonic feedback parameters. We find that
with WL observations alone, it is possible to rule out decay lifetimes smaller
than $\tau=75$ Gyr (at 95 percent CL) for the case that all DM is unstable.
This constraint improves to $\tau=182$ Gyr if the WL data is combined with CMB
priors from the Planck satellite and to $\tau=275$ Gyr if we further assume
baryonic feedback to be fully constrained by upcoming Sunyaev-Zeldovich or
X-ray data. The latter shows a factor of 3.2 improvement compared to
constraints from CMB data alone. Regarding the scenario of a strongly decaying
sub-component of dark matter with $\tau\sim 30$ Gyr or lower, it will be
possible to rule out a decaying-to-total fraction of $f>0.49$, $f>0.21$, and
$f>0.13$ (at the 95 percent CL) for the same three scenarios. We conclude that
the upcoming stage-IV WL surveys will allow us to significantly improve current
constraints on the stability of the dark matter sector.
