Two problems:
There is no HTML entity in the code you posted with an id of "address".
The id attribute must be unique for all objects in an HTML document. All of your textarea's share the same id value.
I would get rid of the id attribute altogether, since you don't need it here. Instead, you can pass a reference to the textarea object by using the keyword 'this', e.g.:
<textarea name="address[]" class="address[]" onkeydown="checkCar(this);" onkeyup="checkCar();" ></textarea>
Then, in your checkCar() function, you'd use the object passed as a parameter rather than searching for the object reference via document.getElementById().
Also, this is unrelated to your problem, but you also have invalid class names (the square brackets aren't allowed).
EDIT2: It looks like I may have been slightly incorrect in my ending statement; square brackets can be used if they are properly escaped.