The reasons why Jay Chou did not sue Song Mengjun may be as follows:

1. Insufficient evidence: In legal proceedings, sufficient evidence is needed to prove the defendant's infringement. If Jay Chou cannot provide enough evidence, then he may not win the case.

2. Strategic considerations: Jay Chou may think that prosecuting Song Mengjun is not worth the time and energy. He may have more important things to focus on or have other strategies that would solve the problem more effectively.

3. Possible reconciliation: Jay Chou and Song Mengjun may have reached some kind of reconciliation agreement. They may negotiate privately to resolve disputes to avoid the impact of legal proceedings on both parties' images and careers.

4. Legal restrictions: According to relevant legal provisions, Jay Chou may not be able to sue Song Mengjun. For example, if Song Mengjun's actions did not infringe Jay Chou's copyright or trademark rights, or in some cases, the law may not allow prosecution.

5. Public image consideration: Jay Chou may think that prosecuting Song Mengjun will have a negative impact on his public image. He may want to keep a low profile and avoid attracting too much public attention and controversy due to the lawsuit.

In short, Jay Chou's failure to sue Song Mengjun may be due to various reasons. This may include factors such as evidence, strategy, likelihood of settlement, legal restrictions, and public image. In actual circumstances, Jay Chou's decision may be based on the weighing and consideration of multiple factors.