search-form-ajax-issue-simplified.php

<footer>
  
  <form action="/" method="GET">
    <input type="search" name="search">
    <input type="submit" value="Search">
  </form>
  
  <div class="search-results">
    <?php
    if ( isset ( $_GET['search'] ) ) {
      $search_results = get_posts( ['s' => $_GET['search'] ] );
      foreach ( $search_results as $post ) {
        // Display the search results...
      }
    }
    ?>
  </div>
  
</footer>
<script>
  (function() {
    $('form').on('submit', function() {
      $.ajax({
        url: '/',
        method: 'GET',
        success: function(response) {
          var html = $(response).find('.search-results');
          // Insert the search results...
        }
      })
    });
  })();
</script>