You would need to check in both Tomcat and Application logs.
Try tailing these log files right after it goes down in you are in Linux. 503 error cause can be many things even SSL can cause sometimes. So, log entry is crucial for troubleshooting.
There are only three reasons a log will not show you an error:
If your Jira is stopping for any other reason, there will always be information written into <jira home>/logs/atlassian-jira.log and/or <jira install>/log/catalina.out (and possibly some other files in those directories).
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