This is very dependant on the gatway you have.
Some gateways will create ordinary dns entries from devices host name via dhcp,
allowing device1.localdomain or device1.lan to be resolved through the gateway dns.
some gateways allow changing the localdomain to something else (or add additional ones) thus allowing you to point your browser to whatever domain it is set to eg device1.home.
other gateways might even allow you to assign hosts a specific dns name. (I know ubiquiti can do this)
usually, the gateway dhcp server pushes search domains to dhcp clients in the likes of lan or home so you can lookup eg device1 instead of the whole device1.lan or device1.home respectively.
I know sometimes browsers tend to be "smart" and use the addressbar as a search-box and leaves you searching for "device.localdomain" on your search engine instead of pointing you at the device.
You can get around that by explicitly type http://device1.localdomain or http://device1.
the .local domain is typically resolved with mDNS, and if you have devices that are slow on responding to queries, it can take some time to resolv.
Your search domains should be listed with search keyword in the /etc/resolv.conf file, if you have any.
The dig tool is very good to test with eg dig @192.168.1.1 device1.localdomain will ask the dns server at 192.168.1.1 for device1.localdomain
the dig tool can be installed with sudo apt-get install dnsutils