OK this is going to be interesting to explain.
Cookies could be used. But don't think you wanna travel that road. So we will go the long way!
Assume:
user.php == where user puts in name
userreact.php == something to do with user's name
userreact2.php == the third page
have the form on user.php and have it set as user. Use POST to go to userreact.php... do anything you want there. Now if you do a redirect your url will be userreact2.php?user=$user that will send them to userreact2.php and have the $user set. That would work. Or you could have a hidden field if there was another form on userreact.php and just make it a <input type="Hidden" value="<?php echo"$user"; ?>">. Ok hope I didn't confuse you. Thanks!
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Chad R. Smith