wordpress ajax search posts

I tried to configure search area in wordpress using ajax, so i add this code into functions.php

<?php
// the ajax function
add_action('wp_ajax_data_fetch' , 'data_fetch');
add_action('wp_ajax_nopriv_data_fetch','data_fetch');
function data_fetch(){

    $the_query = new WP_Query( array( 'posts_per_page' => 5, 's' => esc_attr( $_POST['keyword'] ), 'post_type' => 'post' ) );
    if( $the_query->have_posts() ) :
        while( $the_query->have_posts() ): $the_query->the_post(); ?>

            <a href="<?php echo esc_url( post_permalink() ); ?>"><?php the_title();?></a>

        <?php endwhile;
  wp_reset_postdata();  
 else: 
  echo '<h3>No Results Found</h3>';
    endif;
    die();
}

// add the ajax fetch js
add_action( 'wp_footer', 'ajax_fetch' );
function ajax_fetch() {
?>
<script type="text/javascript">

$("input").keyup(function() {

 var keyword = jQuery('#searchInput').val();
 if(keyword == ""){
  jQuery('#datafetch').html("");
 } else {
  jQuery.ajax({
   url: '<?php echo admin_url('admin-ajax.php'); ?>',
   type: 'post',
   data: { action: 'data_fetch', keyword: keyword  },
   success: function(data) {
    jQuery('#datafetch').html( data );
   }
  });
  });
 }

</script>
<?php
}
?>

then in my template page i added

<div class="container ">

  <div class="d-flex justify-content-center ">

      <div class="searchbar">

 <input id="searchInput" class="search_input" type="text" name=""  placeholder="Search...">

   <a href="#" class="search_icon"><i class="fas fa-search"></i></a>

 </div>

 </div>

</div>
<div id="datafetch">
</div>

but nothing appear when i fill the search text , did i miss something ?
thank you for your help