  In this paper, we study the $\sigma$-self-orthogonality of constacyclic codes
of length $p^s$ over the finite commutative chain ring $\mathbb F_{p^m} + u
\mathbb F_{p^m}$, where $u^2=0$ and $\sigma$ is a ring automorphism of $\mathbb
F_{p^m} + u \mathbb F_{p^m}$. First, we obtain the structure of $\sigma$-dual
code of a $\lambda$-constacyclic code of length $p^s$ over $\mathbb F_{p^m} + u
\mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a
$\lambda$-constacyclic code to be $\sigma$-self-orthogonal are provided. In
particular, we determine the $\sigma$-self-dual constacyclic codes of length
$p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Finally, we extend the
results to constacyclic codes of length $2 p^s$.
