I have change the following in order to works fine. @pablocorezzola I miss two things, one is a scroll bar when chat overflows certain height, and the other one is a separator for show the date if it has changed.

$(document).ready(function() { 
    $(".mytext").on("keyup", function(e){
        if ((e.keyCode || e.which) == 13){
            var text = $(this).val();
            if (text !== ""){
                insertChat("me", text);              
                $(this).val('');
            }
        }
    });
});

Thank you very much for your contribution. The problem of the scroll is already solved. I also added a button to send a message