For　the　Cauchy　problem　with　a　kind　of　non-smooth　initial　data　for　weakly　linearly　degenerate　hyperbolic　systems　of　conservation　laws　with　the　linear　damping　term,　we　prove　the　existence　and　uniqueness　of　global　weakly　discontinuous　solution　u　=　u(t,　x)　containing　only　n　weak　discontinuities　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　obtain　the　existence　and　uniqueness　of　global　weakly　discontinuous　solution,　continuous　and　piecewise　C　1　solution　with　discontinuous　first　order　derivatives,　of　the　flow　equations　of　a　model　class　of　fluids　with　viscosity　induced　by　fading　memory.　©　2008　Birkhaeuser.