After a partital hysterectomy, can ovaries stop function?

Ovarian function. If the ovaries left intact are normal, then they will continue to function normally after surgery. However the ovaries will undergo the natural changes of peri-menopause and menopause as the individual ages.
Not generally. Evidence has shown that ovaries fail more often and earlier after a hysterectomy. Not a given and is probably related to the reason for the hys and procedure and difficulty.
Age dependent. If you are close to 50 years of age the ovaries could loose enough blood supply that could make them stop working. On the other hand if you are 30-45 they should continue to function properly.