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.　©　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature　2024.