In this paper we study a new property for the Glimm potential introduced by L. Caravenna [3].This new property enable us to study scalar conservation laws with a particular source term called linear damping.By the operator splitting method joined with the polygonal approximation method introduced by C. Dafermos [4] we shown the well-posedness of the Cauchy problem for scalar conservation laws with linear damping and finally we show that the solution exponentially decays.