This paper is concerned with the Cauchy problem $$u_t=u_{xx} +f(t,u), \,\,\,
x\in\mathbb{R},\,t>0, $$ $$u(0,x)= u_0(x), \,\,\, x\in\mathbb{R},$$ where $f$
is a rather general nonlinearity that is periodic in $t$, and satisfies
$f(\cdot,0)\equiv 0$ and that the corresponding ODE has a positive periodic
solution $p(t)$. Assuming that $u_0$ is front-like, that is, $u_0(x)$ is close
to $p(0)$ for $x\approx -\infty$ and close to $0$ for $x\approx \infty$, we aim
to determine the long-time dynamical behavior of the solution $u(t,x)$ by using
the notion of propagation terrace introduced by Ducrot, Giletti and Matano
(2014)...
We establish the existence and uniqueness of propagating terrace for a
very large class of nonlinearities, and show the convergence of the solution
$u(t,x)$ to the terrace as $t\to\infty$ under various conditions on $f$ or
$u_0$. We first consider the special case where $u_0$ is a Heaviside type
function, and prove the converge result without requiring any non-degeneracy on
$f$. Furthermore, if $u_0$ is more general such that it can be trapped between
two Heaviside type functions, but not necessarily monotone, we show that the
convergence result remains valid under a rather mild non-degeneracy assumption
on $f$. Lastly, in the case where $f$ is a non-degenerate multistable
nonlinearity, we show the global and exponential convergence for a much larger
class of front-like initial data.
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