18.04 Desktop LTS DNS lookup problem

Hi,

I am using te latest Ubuntu Mate Desktop 18.04 LTS release and have issues getting local DNS to work.
In my network I maintain a central router instance that povides DHCP and DNS successfully over many years. The DHCP assigns a valid address and correct DNS information to my above mentioned network client. However DNS resolution does not work for DNS records maintained in my router for my local network.
See here: (local DNS server on .3.1)

uho@Asus:~/Schreibtisch$ systemd-resolve --status
Global
          DNSSEC NTA: 10.in-addr.arpa
                      16.172.in-addr.arpa
                      168.192.in-addr.arpa
                      17.172.in-addr.arpa
                      18.172.in-addr.arpa
                      19.172.in-addr.arpa
                      20.172.in-addr.arpa
                      21.172.in-addr.arpa
                      22.172.in-addr.arpa
                      23.172.in-addr.arpa
                      24.172.in-addr.arpa
                      25.172.in-addr.arpa
                      26.172.in-addr.arpa
                      27.172.in-addr.arpa
                      28.172.in-addr.arpa
                      29.172.in-addr.arpa
                      30.172.in-addr.arpa
                      31.172.in-addr.arpa
                      corp
                      d.f.ip6.arpa
                      home
                      internal
                      intranet
                      lan
                      local
                      private
                      test

Link 2 (wlp2s0)
      Current Scopes: DNS
       LLMNR setting: yes
MulticastDNS setting: no
      DNSSEC setting: no
    DNSSEC supported: no
         DNS Servers: 192.168.3.1
uho@Asus:~/Schreibtisch$

uho@Asus:~/Schreibtisch$ nslookup filou
Server: 127.0.0.53
Address: 127.0.0.53#53

** server can't find filou: SERVFAIL

uho@Asus:~/Schreibtisch$ nslookup filou 192.168.3.1
Server: 192.168.3.1
Address: 192.168.3.1#53

Non-authoritative answer:
Name: filou
Address: 192.168.3.10

uho@Asus:~/Schreibtisch$ nslookup 192.168.3.10
10.3.168.192.in-addr.arpa name = filou.

Authoritative answers can be found from:

uho@Asus:~/Schreibtisch$

The example above shows that DNS forward lookup for "filou" does not work, only reverse lookup works.
The same behavior with explicit DNS setting in network manager.

Any idea what's wrong? To me this looks weirdly broken.

BTW: Old school setting in /etc/resolv.conf works like a charm.

BR
Uwe