Query not working with adding where clause after case statement on multiple columns

I have following query which is not working with WHERE clause.Any help will be appreciated.

UPDATE abc SET col1 = CASE WHEN a1_date < now() THEN "xyz" END, col2 = CASE WHEN a2_date < now() THEN "xyz" END, col3 = CASE WHEN a3_date < now() THEN "xyz" END

WHERE a1_date != NULL AND a2_date != NULL