For $2X2$ systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications we consider the systems of isentropic gas dynamics, with pressure obeying a $gamma$-law , for the physical range $gammage1$ , and also for the “non-physical'' range $0<gamma<1$ , both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case $gamma=1$ , and also analyze the general case when $|gamma-1|$ is small. Further, our main result also establishes the decay of the periodic solution.