Upkeep is the process of caring for something so that it keeps working. Changing your car's oil every 5,000 miles is an important part of its upkeep.
An actor's monthly visits to the salon for highlights show how much he cares about his hair's upkeep, while a city manager's focus might be on the upkeep of roads and bridges. In either case, there's a kind of maintenance or care involved. Sometimes this noun describes something closer to "financial support," like when a divorcing couple argues about who's responsible for the upkeep of their seven poodles.