In some situations, when a cache is no longer able to be maintained by the owner, Geocaching.com, GPSgames.org and other listing sites, will transfer ownership of the cache listing to a geocacher that is willing to take on the responsibility of maintaining it. Normally, the Cache Owner will contact you to ask if you are willing to accept the cache as your own. When and if you agree, the owner will send an adoption request, with a link to the adoption page. There you go through a process of "I am willing to adopt this cache", and then the cache is transferred to you. In the case of geocaching.com, when a cache has been adopted.the cache page will say "a cache by ........(adopted by .......)" . Note that for third adopters on the Cache page it will not list your name, only the previous cache owner and the first owner. The same is for fourth or more time adopters, the cache page will only show the first and second owners.
For more information, contact your listing site.