Let $k$ be a field, and let $A, B$ be commutative noetherian $k$-algebras.
If either $A$ or $B$ is a localization of a finite type $k$-algebra, then clearly $A\otimes_k B$ is noetherian.
Assume $A=B$. If $A\otimes_k A$ is noetherian, is $A$ a localization of a finite type $k$-algebra?
More generally, if $A\otimes_k B$ is noetherian, is at least one of $A,B$ a localization of a finite type $k$-algebra?
I could not find any such examples, but also not proofs of these facts.