As a practical matter, the offensive posting may stop immediately. Once stripped of their belief that they can post without regard to consequences, these people often stop posting bad stuff. Remember, the Internet users we are talking about are not necessarily sophisticated corporate warriors accustomed to the rough and tumble of business or litigation. Many are unsophisticated penny-stock promoters, hackers or disgruntled employees. The mere filing of a lawsuit can scare them straight.
Injunctions barring the posting of future messages are harder to come by.
Judges find themselves restrained by the First Amendment from barring people from communicating on the Internet. It's a tough issue, and it isn't going to go away overnight. Some day, law and technology may work together to make it harder for unidentified people to publish false, damaging messages. Until that happens, victims of such attacks can begin their legal defense with a solid threat and hope that that's all it takes.