not leaving the page after sending an email

So I am trying to get this Ajax type of popup (so far as I've looked for answers, I only found Ajax and JQuery way to do this) thing after sending email through PHP.

I wish to make a div appear on the same page after I click my 'send' button, but PHP immediately drops me on a blank page right after I click it. I wish it to not leave the page.

It's one of the simpliest php contact form script I could stumble upon:

<?php
$name = $_POST['name'];
$email = $_POST['email'];
$message =...
Code (markup):
not leaving the page after sending an email

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