处理GET请求并调用Javascript

User clicks on my site's url with a GET parameter like so:

www.mywebsite.com/index.php?name=Mark

Let's say index.php looks like that:

if(isset($_GET['param'])){
     MyPhpClass::staticFunction();
     exit();
}
include 'index.html';

Let's make index.html look like this:

<html>
<head></head>
<body><div id='greetings'>Hello</div></body>
</html>

What I want is to do is replace Hello with Hi there using the PHP staticFunction. Note: this is just a toy example. In reality I would like to simulate calling a Javascript function from a PHP function either by calling directly or having some global variable accessible for both.

I have tried something like this:

public static function staticFunction(){
     echo "<script type='text/javascript'>$('#greetings').innerHTML = 'Hi there'</script>";
}

But it doesn't seem to work which makes sense. Another thing that I've tried is setting a global variable in PHP and then trying to access it from JavaScript like so:

public static function staticFunction() {
    $GLOBALS['myglobalvariable'] = 'HiThere';
}

And then including a Javascript inside index.html:

var myglobalvariable = <?php echo $GLOBALS['myglobalvariable']; ?>;
if(myglobalvariable == 'HiThere') {
    $('#greetings').innerHTML = 'Hi there';
}

I suspect that I could make it work using one of the solutions I've mentioned (there might be some bugs there) but I would be very grateful for a suggestion for an elegant solution to the problem I described.

I by no means expect a full solution - a push in the right direction will be great. Thanks!