If a person dies, there are people who inherit what the person owned, but also the debt. In some countries one can refuse to accept an inheritance. The relevant court will search for next-of kin. If nobody acepts the inheritance the creditor is left without anything more than what the person left.
If there are any transaction (new contracts) running, but are not in effect yet, those transactions will cease.
In fact if there is a severe legal problem you should consult a lawyer who know about your local legal system and the appropriate action.
If a person is in debt but he is dead, the transaction will stop, right? But if not what should I do? For me, as far as I know, when someone dies, any transaction will stop.
No, a person is dead when the lawyers are done with him.
If contracts are running (rent, utilities, car payments, etc.), they do not automatically stop, nor do debts simply go away; the person assigned to be the estate executor must terminate the contracts and sell off assets to pay any remaining debts.
Only if you don't make a memorial for yourself, or name a wing of a hospital after yourself, or give to some charity in your name. That will only be remembered for so long though then forgotten. Birds will crap all over your memorial & the wing of the Hospital named after you will eventually get old and be torn down or refinished.
Best thing to do is leave it to those who have treated you well and need it and will appreciate it.
For me, you have to write a letter to let them know, like this to prove your concern.