  We investigate the decomposition of Jacobians of superelliptic curves based
on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an
necessary and sufficient condition in terms of $m$ and $n$ for the
decomposition of the Jacobian induced by the automorphisms of the curve.
Moreover, we generalize a construction in \cite{Ya} of a family of
non-hyperelliptic curves $\mathcal X_{r,s} $ and determine arithmetic
conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$
decomposes.
