What should I do if the damaged carpet installed in my house under warrenty but design center did not respond my letters?

You can hire someone to do an independent evaluation of your carpet.

If it was improperly installed, the only option maybe legal.

If the problem is matting/staining/fiber loss then you could try and deal with the carpet mill yourself. The problem with this is, if this carpet is more then a year old, they will have to have all your documented professional cleaning receipts required under most warranties.