Web service : get two major version with one database

I need to upgrade my rest api : http://myserver/myapi/v1 to http:/:myserver/myapi/v2. As I have several mobile applications that use version 1, I have to keep both APIs online for a while. I use a database behind this API.

My question is simple: the database schema for v2 must evolve.

Is it necessary to create a second database for the V2 version (with the problems of synchronization between the base v1-v2)?

If I keep a single base for both versions, what are the best practices for managing both backends? (evolution of tables used in the V1 with a different scheme). I can not find any information on this topic, we only talk about the API remained, not on how to manage the backend !!