We study an infinite-horizon sequential dynamic game where the players are a government and an international terrorist organization. We provide conditions for the existence of equilibria in which the terrorists’ resources are totally destroyed by a government’s strike. Specifically, we study strong eradication equilibria in which the government’s strike annihilates the terrorists’ resources, preventing the terrorists from acting. We also pay attention to weak eradication equilibria in which the terrorists’ resources are destroyed but in which the initial value of the terrorists’ strike is nevertheless positive. We also show the existence of an equilibrium in which war is perpetual between the government and the terrorists. Perpetual war can only coexist with weak eradication equilibria. For these cases, we provide conditions under which the government would be better off in a weak eradication equilibrium.