Understanding about creating a DFA
You cannot make a DFA for this language because its not a Regular language
in fact its not even a context free language so you can't even make a PDA for it

to prove its not regular and not  context free you should use pumping lemma

but you can make a linear bounded automata for this language so therefore its a context sensitive language

hint for the linear bounded automata: put both of the w's in the tape and separate  them with 0, then compare them.

also here's some examples for some of the strings in the language :

00 , 11 , 0101 , 1010 , 110110 ...