We consider time-periodic reaction-diffusion equations
$$
u_t=\Delta u+f(u,t)
$$
on a bounded domain $\Omega$ under Neumann boundary condition.

The asymptotic  behavior of most bounded  solutions of such equations
is   governed by stable periodic solutions.    We address the question
whether   a stable periodic  solution    can be subharmonic, that  is,
whether its  minimal period   can  be larger  than the  period  of the
equation.   While there are no such  solutions on a  convex domain, we
show  that on some nonconvex   domains stable subharmonic solutions
do occur (if the nonlinearity is chosen suitably).