In the Atlassian Web Site in the FAQ section shows this:
Yes. Just sign up for the OnDemand trial and let your Download trial expire. If you need to migrate data, contact us and we'll help.
You have this question too that can help:
And more details here:
We have a download version which will expire soon. It's at Version: 3.12.3 - # 302. We are considering going to Jira on-demand. Is it possible we can get a trial version for on-demand so we can migrate a project or two and try to some understanding of how the customization will look ?
No, you will need to perform an intermediate upgrade to 4.x first (preferably 4.4.5).
Note that you cannot easily migrate specific JIRA projects to OnDemand. You'd want to import everything (all projects) in one go, evaluate it (but still using your original instance for all real work) and then re-import everything if/when you migrate.
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