If it you have estate or spouse, then it will go with them. The company will first try to collect from the estate and if the account is a joint account then any other who is a part of it will have to get with it. And if the debt is solely belonging to the deceased then the company will end up eating the debt if there are not big and enough assets to cover it.
Re: Who is responsible for the debt amount in your credit card after your death?
After death as soon as possible get at least a dozen copies of death certificates. Order a copy of your credit card as well as yours card. Notify the credit agencies of the death. They will eventually get the notification through other sources.