  We show that the smoothed particle hydrodynamics (SPH) method, used with
individual time-steps in the way described in the literature, cannot handle
strong explosion problems correctly. In the individual time-step scheme,
particles determine their time-steps essentially from a local Courant
condition. Thus they cannot respond to a strong shock, if the pre-shock
timescale is too long compared to the shock timescale. This problem is not
severe in SPH simulations of galaxy formation with a temperature cutoff in the
cooling function at $10^4 {\rm K}$, while it is very dangerous for simulations
in which the multiphase nature of the interstellar medium under $10^4 {\rm K}$
is taken into account. A solution for this problem is to introduce a time-step
limiter which reduces the time-step of a particle if it is too long compared to
the time-steps of its neighbor particles. Thus this kind of time-step
constraint is essential for the correct treatment of explosions in
high-resolution SPH simulations with individual time-steps.
