How to check what happend. I add that docker logs -f container_name | container id neither does work. The solution for me is to remove the actual container (condition that it has some persistent storage) and run new one with entrypoint parameter replaced:

Was:

docker run  -v /dockers_persistent/data/database_1:/var/lib/mysql -v \
MYSQL_ROOT_PASSWORD='*******' -d f44353327cb6

To be:

docker run -it --entrypoint /bin/bash  \
-v /dockers_persistent/data/database_1:/var/lib/mysql -v \
MYSQL_ROOT_PASSWORD='*******' -d f44353327cb6

Now, you can execute the adequate app – in my case it was mysqld.