If user doesn't complete the goal after starting it(which is captured in eVar1) with in the same visit or in the next 3/4 visits, then also eVar value will persist or it will expire? if expire then when it will?
I assume you mean that your eVar's Expiration setting has been set to be some success event, e.g. "Purchase". If the user never does something to cause that success event to fire, e.g. he never makes a purchase, then the eVar will continue to persist forever and ever until he finally causes the success event to fire, i.e. makes a purchase.