However. When the new form is then submitted the page is redisplayed with the results and the div element is again hidden.
I have a JSP variable I can access to find out it the user has choosen a form in the <select><option>...
It is called you have to come up with a solution that shows the div. Easy solution is to have the serverside code set the values if it knows it should be displayed. So you spit out the correct HTML based on the situation you are in.