We consider a noise-limited wireless sensor network that consists of battery-operated nodes which can route information to a mobile sink in a multi-hop fashion. The problem of maximizing the network's lifetime, defined as the period of time during which the network can route a feasible flow to each sink location subject to power/energy constraints, is cast into a linear program, reduced into a simpler equivalent form and solved via dual decomposition. The unknowns are the sink sojourn times and the routing flow vector for each sink location. The presence of a mobile sink presents new challenges but the problem structure can still be exploited to find the optimal solution. A distributed algorithm based on the subgradient method and using the sink as leader is proposed and its performance is evaluated through simulation for random networks. The algorithm's requirements in memory are also provided.