Above comments withstanding, here it goes:

Change this:

if ($query->param()) {
     $search_name = $query->param('search_name');
...
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into:

if( $search_name = $query->param('search_name') ) {
...
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If the param('search_name') is empty/fails, it will return a blank string, '', which perl treats as a false value, i.e., if condition fails. Same effect as before, a little clearer.

Why are you doing <BR> and \n here? \n won't have any real effects on the html(usually), so it's not needed. Unless, I presume, you need to edit it, and this is for 'beautification purposes'. If you really want/need to line breaks, try <p>. This is where I'm talking about:

print "$name<BR>\n$email<BR>\n$comments<BR>\n<HR>\n";
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Major Recommendation: Look up CGI.pm's HTML handling abilities! Especially its abilities with forms. It'll make your pages a lot less error prone, and the CGI a lot more bearable/easier to read for larger examples. For example, <title>this is the title</title> can be re-written, assuming $q is a cgi object, as $q->title("This is the title"). CGI will automatically add an /title, or end whatever tag, as needed. Look into it.