Há outros usos para o phrasal verb give up, tais como...
Give “somebody” up (to stop hoping for or expecting somebody to arrive) significa desistir de esperar alguém:
“I’d given her up when she arrived with that silly excuse.”
Give “somebody” up (to stop hoping for or expecting somebody to get better after illness) significa “desenganar” alguém que está muito doente, como dizemos em linguagem popular:
“The doctor had given her up but she made a remarkable recovery!”
Give “oneself” up “to somebody” (to ofer oneself to be captured) significa se entregar, talvez como nos exemplos citados por "efrance":
“After a week on the run, he gave himself up to the troopers.” / “After all he did to seduce her she gave herself up to him.”
Finalmente, give up “on somebody” (to stop trying to support or help somebody) significa desistir de ajudar ou apoiar alguém:
“She gave up on him after the third time he got arrested.”
Acho que isso é tudo sobre give up.
Abraço a todos!