There are many people who comment on other posts or on the Facebook page, twitter, Yahoo answers and other kind of sites about how to kill Pou, about if Pou will die if they fail to take care of it or how much you take this to a happier life.
Sorry to disappoint many, but Pou can't die. There is no way to kill it. If you try to not care for him in any way, trying to kill him by hunger, falling ill or even that he becomes crazy by not sleeping... leave it, you are wasting your time. Pou will be always there waiting to you animes to return to care for him.
The only way to rid of your favorite pet so far is to uninstall the app from your smartphone. But in this way Pou just disappears from your life. In addition to it, if you decide to return to the app later, only by entering your email address with which you registered a name for your Pou, this returns to your terminal as you left it.
So you know, it is impossible to kill Pou. Anyway I think that this little alien don't deserve to die. Keep taking care of it. :)