run of of memory in JTextArea

As my program keeps running and append text to the JTextArea, it soon throws "out of memory" exception. So how do i tackle this problem without setting the text to "". Coz i still want to retain the history, maybe just delete half.

I'm not positive about this because I've never ran out of memory, but maybe you should try a JTextPane or JEditorPane instead of a JTextArea, which is pretty limited.