Hi Majid,
Thank you for the details!
I have tried a couple of products, but all seem to have the default search bar instead of ajax search pro. It does not seem like a cache issue, but I suspect the product pages may use different headers as the home/search page.
If you are using a theme builder of some sort, check the product page headers template, it is very likely different. If it was the same, te same output would be displayed on the home and product pages.