  We study the existence of unstable classical solutions of the
Rayleigh--Taylor instability problem (abbr. RT problem) of an inhomogeneous
incompressible viscous fluid in a bounded domain. We find that, by using an
existence theory of (steady) Stokes problem and an iterative technique, the
initial data of classical solutions of the linearized RT problem can be
modified to new initial data, which can generate local-in-time classical
solutions of the RT problem, and are close to the original initial data. Thus,
we can use a classical bootstrap instability method to further obtain classical
solutions of (nonlinear) RT instability based on the ones of linear RT
instability.
