It　is　well-known　that　the　Rayleigh–Taylor　(RT)　problem　of　an　inhomogeneous　viscoelastic　fluid　defined　on　a　bounded　domain　is　unstable,　if　the　elasticity　coefficient　κ　is　less　than　some　threshold　κ　C　.　In　this　paper,　we　rigorously　prove　the　existence　of　a　unique　unstable　strong　solution　in　the　sense　of　L　1　-norm　for　the　RT　problem　in　Lagrangian　coordinates　based　on　a　bootstrap　instability　method,　when　κ<κ　C　.　Applying　an　inverse　transformation　of　Lagrangian　coordinates　to　the　obtained　unstable　solution,　we　can　further　get　a　unique　unstable　solution　for　the　RT　problem　of　inhomogeneous　viscoelastic　fluids　in　Eulerian　coordinates.　©　2019　Elsevier　Inc.