Adobe make lots of stuff. When you say "Adobe" do you mean the Adobe Acrobat Reader DC app?
The FREE App for iPhone & iPad, the icon says "Adobe Acrobat", I think it is Adobe Acrobat Reader....Thanks.
Perhaps someone can move this to the Adobe Reader mobile forums where the experts hang out.
Once you accept a document via AirDrop and opening it in Acrobat Reader for iOS, the document will be copied to your iPhone/iPad ('Local').
You cannot transfer documents from Recent to Local. Recent just displays recently viewed local and cloud documents.
You can switch the file location to Local to find the document.
Hope this helps.
Please let us know if you have additional questions.
Thank you, it works.