It's nearly 100% certain that the failing application is defective and from the sound of it, it probably has a slow resource leak. Tomcat is very reliable, and when it does fail, it is almost always going to fail for all apps, not just one of them.
Slow resource leaks can take a certain amount of detective work. They often occur because of things like people forgetting to close files (or thinking that they don't have to), database exception handlers that don't close connections, cleanup services that don't actually clean up - causing the number of things being "cleaned" to get bigger and bigger". Stuff like that.
You'll probably need both Tomcat logs and application logs. I would also interview some of the users and see if they start seeing problems (slow response, error pages) that build up the longer the application has been running. That may give you an idea of where to investigate further.
When it comes to destroying a civilization, gas chambers cannot hold a candle to echo chambers.