It　is　proven　that　a　class　of　the　generalized　Riemann　problem　for　quasilinear　hyperbolic　systems　of　conservation　laws　with　the　uniform　damping　term　admits　a　unique　global　piecewise　C-1　solution　u　=　u　(t,　x)　containing　only　n　shock　waves　with　small　amplitude　on　t　＞=　0　and　this　solution　possesses　a　global　structure　similar　to　that　of　the　similarity　solution　u　=　U　(x/t)　of　the　corresponding　homogeneous　Riemann　problem.　As　an　application　of　our　result,　we　prove　the　existence　of　global　shock　solutions,　piecewise　continuous　and　piecewise　smooth　solution　with　shock　discontinuities,　of　the　flow　equations　of　a　model　class　of　fluids　with　viscosity　induced　by　fading　memory　with　a　single　jump　initial　data.　We　also　give　an　example　to　show　that　the　uniform　damping　mechanism　is　not　strong　enough　to　prevent　the　formation　of　shock　waves.　(c)　2006　Elsevier　Inc.　All　rights　reserved.