Can you please mention the firmware version of 7937?
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Firmware 1.4(5) is available for the 7937. The release notes don't mention your problem, but there are other DHCP fixes in the firmware.
I haven't seen anything like this with our 7937s. Have you Wiresharked the phone, to be 100% sure what the phone is getting via DHCP is what you think it should be getting?
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You probably want to try firmware version 1.4.4ES75 which contains the fix for this DHCP bug where the phone does not properly handle DHCP lease times greater than 7 days, https://tools.cisco.com/bugsearch/bug/CSCtr56885. In order to get this firmware version please open a TAC case and reference this defect.