Instead of local IP addresses by host name, I get the IP DNS server?

Question

Domain Name System

Hello everyone)
There is a Linksys e4200 router, an Internet is plugged into it (directly without any PPTP) and 2 local machines PC1 and PC2 with addresses 192.168.1.XXX and 192.168.1.YYY.
The problem is that when I ping PC2 from PC1 by name, I get back not 192.168.1.YYY but the provider's DNS server address in general.
Any ideas how to fix?

Answer 1

What does traceroute say?
I'm not talking about the fact that it would be nice to know how addresses are assigned to local machines, which network the provider has and which routing table is on the router.

Answer 2

I'm sorry, I wrote in the wrong place ...))