In legal language, we use "support obligation" to refer to the duty that parents and children have to provide mutual "support", namely to give one another the means of subsistence.
In common language, the support obligation generally refers to the support payment paid for a child or to a former spouse following a separation, a divorce or the dissolution of a civil union.
Someone cannot, by will, terminate support payments. His estate will assume this responsability for a certain period of time.
Moreover, there are resources that can help an individual who has a judgment granting support enforce that judgment even when the party required to pay support resides abroad.