DB2 Express edition is (actually was, as it's not available anymore) not free; DB2 Express-C is the free edition.
If you license a paid edition of DB2 its license file will be available for download from the usual source, IBM Passport Advantage. DB2 Express-C license is not available separately, it can only be had by downloading and installing the Express-C package, at which point you might as well just use it for your POC.
To install a license file you'd use the
db2licm (license manager) utility by running this as the target instance owner:
db2licm -a <path to license file>
You may want to double-check the demo license agreement though -- it probably says that you must remove the demo installation after its license expiration, which would mean that installing the Express-C license into it, though it might be technically possible, would violate the agreement.
Update: According to the documentation you can continue using the installed product after the trial license expiration by registering a valid license file; it looks like adding the Express-C license should be OK. Note though that if you have used in the trial any of the features unavailable in the Express-C edition, your installation may behave erratically or even fail to start.