  In this paper, we investigate the stabilization of a locally coupled wave
equations with local viscoelastic damping of past history type acting only in
one equation via non smooth coefficients. First, using a general criteria of
Arendt-Batty, we prove the strong stability of our system. Second, using a
frequency domain approach combined with the multiplier method, we establish the
exponential stability of the solution if and only if the two waves have the
same speed of propagation. In case of different speed propagation, we prove
that the energy of our system decays polynomially with rate 1/t. Finally, we
show the lack of exponential stability if the speeds of wave propagation are
different.
