cannot connect ESP8266 web server after some time
The esp8266 enters modem sleep in STA mode without WiFi traffic. You can turn off the sleep mode with WiFi.setSleepMode. The possible parameters are WIFI_NONE_SLEEP, WIFI_LIGHT_SLEEP or WIFI_MODEM_SLEEP.

After some longer time the servers on esp8266 with arduino core usually get inaccessible even it the esp8266 connects as client to other servers or still sends UDP messages. It is a known issue with some possible workaround documented.