Yes. According to CBP, when your ESTA is about to expire, an expiration notice can be sent to the email address provided in your application.
This is why it is important to keep the email address on your application accurate and accessible.
However, before traveling, it is still a good idea to check your ESTA status yourself instead of relying only on email.
If your authorization has already expired, you must submit a new application rather than extend the old one, so please apply fo ESTA before your departure.