Hi Pipvan,
I had this happen to me as well, and it was on a set of radial tyres. I replaced the tyres with 'proper' whitewalls. However please beware of what type and where they originate from if you go down this route. My first set of actual whitewalls were manufactured in South America, and after a few years the white part was covered in little splits, similar to what you see on an older, perished tyre.
Happily the tyre wasn't compromised as the white part was vulcanised onto the actual sidewall of the tyre. When I opened the splits up, the sidewall of the tyre was fine, although it really spoiled the look.
In the end, I bit the bullet and bought Firestone 'Wide Whites', very expensive, but excellent quality, and totally trouble free.
(To keep them clean, I used a liquid called 'Blech-White', brilliant, although there will be other products available.)
Good Luck!
Gavin Doole.