Unknown IMAP connections

My mail server (running on a Intel Xserve on Mac OS X 10.4.11) has started to behave strangely. Basically, I get tons of unknown connections to my IMAP server, that looks like this:

So, tons of connections from IP-addresses only, no user names. This fills up my available slots and the server becomes unresponsive to new connections. Instead of lifting the limits, I would very much rather find out what the problem is and if there is anything I can do to help this.

When this happens, the /var/log/mailaccess.log file starts to fill with items such as:

localhost imap[2207]: auxpropfunc error generic failure

localhost imap[2207]: DBERROR: opening /var/imap/mailboxes.db: Cannot allocate memory

localhost imap[2207]: DBERROR: opening /var/imap/mailboxes.db: cyrusdb error

localhost imap[2207]: Fatal error: can't read mailboxes file

But as far as I can determine, the RAM is sufficient (the server has 2GB and rus only mail) and there is plenty of disk mounted on /, so

The IP's that are for unknown users all belong to Tre - a known swedish mobile operator, so I am guessing that this has more to do with a faulty mobiile client of my users than an outright attack