Global search in product detail page (mobile) necessary?

I'm currently redesigning a product page for a e-commerce mobile application.

The application is pretty compact: it currently consists of having one "home" page with all products in it. Users can scroll at the list of products, browse and click on a product they want. There's also a global search bar at the top navigation bar of the "home" page.

Once a user clicks on a product and enters a product detail page, from there on the user can either
1) see related products (enter another product page) , 2) see recommended products (enter another page), or 3) click on recommended keywords (and enter the search page.)

In order to go back to the "home" page, the user has to click back once, if not multiple times. I found this process frustrating for the user. If the user surfs through multiple products, it meant the user had to go back multiple times.

My question is:

1) Would it be a nice user experience to add the global search button the top navigation bar of the product page?

2) Would it also be necessary to add a short cut to the "Home" page on the top navigation bar of the product page?

3) Or, would it be better to add a bottom navigation bar at the product page, with Home, Search, and other menus.