iPhone connects to WiFi but drops Internet connection
This is abnormal behaviour, and therefore probably a bug in either your wireless router or iOS, more likely your router. It seems to assign your iPhone a static IP after some time for no visible reason (although you haven't used a setting to tell your router to do that).

A list of what you should try, try one after another if it still does not work:


Check for firmware updates for your wireless router. That rules out any known issue to the manufacturer, where fixes are already available.
Reset your wireless router to factory defaults, as that rules out any issues with configuration (intended or unintended).
Tell your iPhone to "forget" your wireless network, and connect to it again, supplying the password. That way, you're ruling out any issues with configuration files with your particular network.
Make an encrypted backup of your iPhone using iTunes if you want to be able to rollback any of the following changes.
Reset your iPhone's network settings (In Settings > General > Reset > Reset Network Settings). You'll have to reconnect to all wireless networks again, as it also deletes passwords, but it makes sure that no issues exist in configuration files regarding general network configuration.
Do a factory restore on your iPhone. If it works now, try restoring the backup and check if the issue is still fixed.
If it still does not help, you might consider calling Apple support (although there is not much more they can do).