If the DNS name is not properly registered to the DNS server through DHCP server for linux machine, which fault is this ?
from the DHCP console it is blank with no hostname ?
If anyone know how to fix this, that be great.
I would not call it fault because there are reasons for one or the other behavior.
In your case you should be able to adapt the dhclient config (IIRC /etc/dhcp/dhclient.conf) on your Debian machine.
You'll have to add a line like the one below
There are two ways for DNS registration of DHCP leases to occur:
Both are valid approaches, though the second one requires Dynamic DNS registration support on the part of the client.
In your case it's impossible to say who is doing the registering. One or the other is not registering the hostname of the machine that received the lease.
As other posters say, it could be at either end.
One option is to use a lease reservation in the DHCP scope based the MAC address of the client machine. Create a static DNS entry for the reserved IP. Document why and how this is done, it will need to be recreated if the DHCP server changes or you need to add another host name.