Quality is delighting customers
Your Heading is :- Compatibility testing and further your question is something else... so what do you mean to ask..??
anyway :- as far as testing any application with different browsers is concerned , You need to understand that Cross browser testing has its main focus on User Interface then Functionality and Security ... so for User interface issues you need not to test with different test cases and testing an application with different browsers has nothing to do with different test cases...!!
Functional Test Cases not required for Cross Browser Testing. Test your application in different browsers.
Note: The Cross browser testing is mainly focused GUI.
Based on my knowledge no need of writing separate test cases for different browsers. But you must test in different browsers because some cases CSS, Ajax .... etc behave differently in different browsers.
Testing the functionality of the application with different versions of s/w or networks or platforms
The focus of the compatibility testing is the critical and major functionality of the application
It classifies in to 2 types
Forward compatibility: Testing the functionality of the application with higher versions of the s/w with respect to base version.
Backward compatibility: Testing the functionality of the application with lower versions of the s/w with respect to base version.
As per question, u are talking abt Cross browser testing, we will be referring to same test cases and we will be checking if the application is behaving same way in all the browsers.
Regarding compatibility testing, Madhurima has explained .