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　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　C1　solution　with　discontinuous　first　order　derivatives,　of　the　flow　equations　of　a　model　class　of　fluids　with　viscosity　induced　by　fading　memory.