Ajax call does not activate callback function
Since you’re serializing the form, as ajax data, you could be missing the action part from your form: <input type=”hidden” name=”action” value=”bda_alert”> so your ajax request would contain /wp-admin/admin-ajax.php?action=bda_alert&… that should invoke the wp_ajax_bda_alert or wp_ajax_nopriv_bda_alert action callback.