Constacyclic codes of length $p^sn$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R=\mathbb{F}_{p^m}[u]/\langle u^2\rangle=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$, where $p$ is a prime and $m$ is a positive integer. For any $\lambda\in \mathbb{F}_{p^m}^{\times}$, an explicit representation for all distinct $\lambda$-constacyclic codes over $R$ of length $p^sn$ is given by a canonical form decomposition for each code, where $s$ and $n$ are positive integers satisfying ${\rm gcd}(p,n)=1$. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct negacyclic codes and their dual codes of length $p^sn$ over $R$ are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over $\mathbb{F}_5+u\mathbb{F}_5$ of length $2\cdot 5^s\cdot 3^t$ are listed for any positive integer $t$.