"Dylan and Carys have been so amazing and supportive throughout all of this," Douglas, 66, tells PEOPLE of his young children.
Douglas, who stepped out in N.Y.C. Tuesday night for a Variety awards-season screening of Wall Street: Money Never Sleeps and told reporters "I feel good," is beginning to regain his strength after undergoing radiation and chemotherapy.
A friend adds that he has also leaned on his wife of 10 years: "[Catherine] is a strong woman and very practical. There's no two ways about it – that's the Welsh in her. Neither one of them are people who would wallow; they just deal with what has to be done."
For much more on Douglas's recovery, pick up this week's issue of PEOPLE, on newsstands Friday