[WiFi problems topic

I just got Xperia Z1 and I have came accross some WIFI problems... The WIFI on the phone works, along with the mobile data, but wifi works only untill the screen switches off, when it switches of, it still is connceted but will not download anything. It also often happens that in my wifi signal icon I can see only the up arrow(uploading data to wifi) while the down arrow(download) is black. The wifi signal is excellent, the stamina mod is not on, only the low battery mode(when battery drops low), phone is charged. I updated it today to 14.4.A.0.157
Has anybody encountered this problem?
By the way, over mobile data, it works flawlessly