$db = JFactory::getDbo();
$query = $db->getQuery(true);

$query->select('a.*,b.category_name')
        ->from('#__carlist AS a')
        ->innerJoin('#__categories AS b ON b.id = a.category_id');

if ($name != '') {
$query->where("a.db_name like '%" . $name . "%'");
}

$db->setQuery($query);
$rows =  $db->loadObjectList();

a.db_name like '%hello me%'  // in the end should be like that

Like '%hello%me%'

Like '%hello%' or like '%me%'

MATCH( a.db_name) AGAINST ('hello me' in BOOLEAN MODE)