I doubt it is a dos attack hitting the search.php page.
Running a search will require resources and if maxed out, any extra connection will give the max connection error. If your site is even just fairly busy you can get hundreds of these errors once you peak.
Does this happen at the same daily? maybe your site is running a cron task at that time or doing a back up which is also using resources.
If you are on a shared host your host will most likely spot a real ddos attack long before you will. If on a dedicated that is a different problem all together.
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