I don't see any reason to restrict your cache to a specific user. Isn't it likely that a day when one person searches for "Yasser Arafat" someone else will too? Maybe not, but if this is a feature of your application how about this solution?
Every search for a name can bre recorded. The first step when performing a new search is to check if the cache helps. Periodically, a cron job removes the least recently used results.
This will work better if what you cache is not the result of a complex query - which is less likely to be reusable - but the result of an atomic search, with a single match criterion.
It will be like having a partial index into your database.
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