Hi!
Unfortunately there is no option for that on the back-end. However, I think the quickest solution in this case would be, to create 2 search instances, one for desktop, one for mobile. (you can make a duplicate of this one and change it)
Then on the mobile search instance, you could change the settings state to hidden by default.
To change the search visibility, you can do that here: https://i.imgur.com/gbLanll.png
