Sending email with php and ajax

Question

I am trying to send email from my page but it does not return any message to me if it was successful or not, I am using php along with bootstrapvalidation and ajax to forward the information to my php page.

JS

// Contact Form Scripts

$(function() {

$("#contactForm input,#contactForm textarea").jqBootstrapValidation({
    preventSubmit: true,
    submitError: function($form, event, errors) {
        // additional error messages or events
    },
    submitSuccess: function($form, event) {
        event.preventDefault(); // prevent default submit behaviour
        // get values from FORM
        var name = $("input#name").val();
        var email = $("input#email").val();
        var assunto = $("input#assunto").val();
        var message = $("textarea#message").val();
        var firstName = name; // For Success/Failure Message
        // Check for white space in name for Success/Fail message
        if (firstName.indexOf(' ') >= 0) {
            firstName = name.split(' ').slice(0, -1).join(' ');
        }
        $.ajax({
            url: "././mail/contact_me.php",
            type: "POST",
            data: {
                name: name,
                assunto:assunto,
                email: email,
                message: message
            },
            cache: false,
            success: function() {
                // Success message
                $('#success').html("<div class='alert alert-success'>");
                $('#success > .alert-success').html("<button type='button' class='close' data-dismiss='alert' aria-hidden='true'>&times;")
                    .append("</button>");
                $('#success > .alert-success')
                    .append("<strong>Sua mensagem foi enviada, em breve entraremos em contato.. </strong>");
                $('#success > .alert-success')
                    .append('</div>');

                //clear all fields
                $('#contactForm').trigger("reset");
            },
            error: function() {

                // Fail message
                $('#success').html("<div class='alert alert-danger'>");
                $('#success > .alert-danger').html("<button type='button' class='close' data-dismiss='alert' aria-hidden='true'>&times;")
                    .append("</button>");
                $('#success > .alert-danger').append("<strong>Sorry " + firstName + ", it seems that my mail server is not responding. Please try again later!");
                $('#success > .alert-danger').append('</div>');
                //clear all fields
                $('#contactForm').trigger("reset");
            },
        });
    },
    filter: function() {
        return $(this).is(":visible");
    },
});

$("a[data-toggle=\"tab\"]").click(function(e) {
    e.preventDefault();
    $(this).tab("show");
});
});


  /*When clicking on Full hide fail/success boxes */
  $('#name').focus(function() {
  $('#success').html('');
  });

HTML

<form action="mail/contact_me.php" method="post" name="sentMessage" id="contactForm" novalidate>

                        <div class="row">
                            <div class="form-group col-md-6">
                                <input type="text" class="form-control" placeholder="Nome *" id="name" required data-validation-required-message="Por favor, informe seu nome.">
                                <p class="help-block text-danger"></p>
                            </div>

                            <div class="form-group col-md-6">
                                <input type="email" class="form-control" placeholder="E-mail *" id="email" required data-validation-required-message="Por favor, informe seu e-mail.">
                                <p class="help-block text-danger"></p>
                            </div>

                            <div class="form-group col-md-12">
                                <input type="text" name="assunto" id="assunto" class="form-control" placeholder="ASSUNTO*" required data-validation-required-message="Por favor, informe um assunto.">
                                <p class="help-block text-danger"></p>
                            </div>


                            <div class="col-md-12">
                                <div class="form-group">
                                    <textarea class="form-control" rows="5" placeholder="Mensagem *" id="message" required data-validation-required-message="Por favor, insira uma mensagem."></textarea>
                                    <p class="help-block text-danger"></p>
                                </div>
                            </div>

                            <div class="clearfix"></div>

                            <div class="form-group">
                                <div id="success">teste</div>
                                <button type="submit" class="btn btn-primary pull-right">ENVIAR MENSAGEM</button>
                            </div>

                        </div>
                    </form>

PHP

        <?php

        /*Checa se os campos estao vazios*/
        if (empty($_POST['name']) ||
                empty($_POST['email']) ||
                empty($_POST['assunto']) ||
                empty($_POST['message']) ||
                !filter_var($_POST['email'], FILTER_VALIDATE_EMAIL)) {
            echo "No arguments Provided!";
            return false;
        }

        $name = strip_tags(htmlspecialchars($_POST['name']));
        $email_address = strip_tags(htmlspecialchars($_POST['email']));
        $assunto = strip_tags(htmlspecialchars($_POST['assunto']));
        $message = strip_tags(htmlspecialchars($_POST['message']));

        /*Configurações de envio de Email*/

        $to = '[email protected]'; /*Email de Recebimento*/
        $email_subject = "Formulario de Contato"; /*Assunto do Email*/

        $email_body = "Nome: $name\n"
                . "Email: $email_address\n"
                . "Assunto: $assunto\n\n"
                . "Mensagem:\n$message";
        $headers = "From: " . $email_address . "\n"; /*Email de quem enviou*/
        $headers .= "Reply-To: $email_address";
        mail($to, $email_subject, $email_body, $headers);
        return true;