  This paper is concerned with the asymptotic behavior of solutions of time
periodic reaction-diffusion equation \begin{equation*}\label{aaa} \begin{cases}
u_{t}(x,t)=u_{xx}(x,t)+f(t,u(x,t)),\quad \,\,\forall x\in\mathbb{R},\,t>0,\\
u(x,0)=u_{0}(x), \quad \quad\quad\quad\quad\quad\quad\quad\quad \forall
x\in\mathbb{R}, \end{cases} \end{equation*} where $u_{0}(x)$ is the Heaviside
type initial function and $f(t,u)$ satisfies $f(T+t,u)=f(t,u)$. Under certain
conditions, we prove that there exists a minimal propagating terrace (a family
of pulsating traveling fronts) in some specific sense and the solution of the
above equation converges to the minimal propagating terrace.
