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In this paper, we study the $\sigma$-self-orthogonality of constacyclic codesof 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 $\mathbbF_{p^m} + u \mathbb F_{p^m}$. First, we obtain the structure of $\sigma$-dualcode 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. Inparticular, 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 theresults to constacyclic codes of length $2 p^s$.

$\sigma$-self-orthogonal constacyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$