Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $G$ that are not isomorphic to $\mathbb{G}_a^n$ over $K$, but are isomorphic to $\mathbb{G}_a^n$ after base changing to some finite purely inseparable extension of $K$.
Does there exist a smooth equivariant compactification of a non-trivial form of $\mathbb{G}_a^n$ over a non-perfect field of positive characteristic such that the boundary divisor is geometrically reduced?
The following is a non-example. Suppose that $K=\mathbb{F}_2 [t]$. Let $G$ be the subgroup of $\mathbb{G}_a^2$ given by the curve $x^2+t y^2=y$. Every non-trivial form of $\mathbb{G}_a$ with infinitely many $K$-points is isomorphic to $G$ (p.68, [1]). One can see that $\mathbb{P}^1$ is a smooth equivariant compactification of $G$. The boundary divisor is an inseparable point of degree $2$, which is not geometrically reduced. As the regular completion of a curve is unique up to isomorphism, we conclude that there exists no compactification where the boundary divisor is geometrically reduced. One can prove the same for any non-trivial form $\mathbb{G}_a$ with finitely many points as well.
[1] p.68, J. Oesterle, Nombres de Tamagawa et groupes unipotents en charactérstique p, Invent. Math. 78, No. 1, 13–88 (1984).
