Per LinkedIn's Share on LinkedIn documentation, the "Source" is supposed to represent
the source of the content (e.g. your website or application name). That said, it may block/prohibit a url as a value (but I doubt that). There's a couple possible reasons why the source information isn't being populated from AddThis' side.
1) AddThis platform's LinkedIn Share button code is looking for a setting/attribute from your AddThis account where you setup the button. You may need to register the domain in AddThis where you configured the button.
2) AddThis code is looking for missing metatag data. Try adding metatags in your site-wide templates for the application-name, author, creator, and publisher. Refer to https://developer.mozilla.org/en-US/docs/Web/HTML/Element/meta
If both of those approaches hit a dead-end, I'd recommend you submit a help ticket with AddThis. Their developer / API documentation doesn't provide much insight on how they determine/populate the source parameter for the share link.