Column comparison between rows of the same group used in Partition By clause

I want to compare same columns of rows falling into the same group created by partition by clause in SQL. My table has below data:

Id  Serial_Number   Last_update_date    Flag
1   3344            07-05-2024          N
2   3344            02-05-2024          N
3   9098            20-02-2024          Y
4   9098            20-02-2024          Y
5   2323            07-03-2023          Y
6   2323            17-10-2023          Y
7   2323            17-10-2023          Y
8   1894            10-07-2022          N
9   1894            20-02-2024          N
10  1894            02-05-2024          N

In the above data, grouping should be done by Serial_Number column and should check if any 2 dates within a serial_number are same, the flag for all the group members should be Y, otherwise all should be N. I tried using count(distinct last_update_date) over partition by serial_number but it is not working.

with cte as (
  select a.*, count(distinct last_update_date) 
                   over (partition by serial_number) as cnt
  from my_table a
)
select * from cte 

If any 2 dates in partition is equal then max(dense_rank()) < count(*) for this partition.

with cte as (
  select a.*
    ,dense_rank()over (partition by serial_number order by last_update_date) as rnk
  from my_table a
)
select * 
  ,case when max(rnk)over(partition by serial_number)
               < count(*)over(partition by serial_number) 
       then 'Y' 
   else 'N' 
   end flag
from cte 
order by id

demo