\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 185, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/185\hfil Spreading solutions in time-periodic environment] {Spreading solutions for a reaction diffusion equation with free boundaries in \\ time-periodic environment} \author[F. Li, J. Lu \hfil EJDE-2018/185\hfilneg] {Fang Li, Junfan Lu} \address{Fang Li \newline Mathematics \& Science College, Shanghai Normal University, Shanghai 200234, China} \email{lifwx@shnu.edu.cn} \address{Junfan Lu (corresponding author)\newline School of Mathematical Sciences, Tongji University, Shanghai 200092, China} \email{1410541@tongji.edu.cn} \thanks{Submitted December 20, 2017. Published November 15, 2018.} \subjclass[2010]{35B40, 35R35, 35K55, 92B05} \keywords{Spreading phenomenon; free boundary; time-periodic environment} \begin{abstract} In this article, we consider a reaction diffusion equation with free boundaries in a time-periodic environment. Such models can be used to describe the spreading of a new or invasive species over a one-dimensional habitat, with the free boundaries representing the expanding fronts. We study an equation with a general time-periodic nonlinearity, and present some sufficient conditions for spreading phenomena. We also use time-periodic semi-waves to characterize the spreading solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} In this article, we study the spreading phenomena of the time-periodic reaction diffusion equation with free boundaries, \begin{equation}\label{main-eq} \begin{gathered} u_t=u_{xx}+f(t,u),\quad t>0,\; g(t)0,\\ u(t,g(t))=0, g'(t)=-\mu u_x(t,g(t)), \quad t>0,\\ -g(0)=h(0)=h_0, u(0,x)=u_0(x),\quad -h_0\leq x\leq h_0, \end{gathered} \end{equation} where $\mu$ and $h_0$ are given positive constants, $u_0$ is a nonnegative function with support in $[-h_0,h_0]$, $x=g(t)$ and $x=h(t)$ are the moving boundaries to be determined together with $u(t,x)$. Moreover, for some $T>0$, $\gamma\in (0,1)$ and some $\alpha(t)\in C(\mathbb{R})$ ($T$-periodic and $\alpha^0 :=\max\alpha(t)>\alpha_0 :=\min\alpha(t)>0$), the function $f$ is a general nonlinearity satisfying the assumption \begin{itemize} \item[(H1)] $f(t,u)\in C^{\gamma/2,1}_{\rm loc}([0,T]\times\mathbb{R})$ is $T$-periodic in $t$, $f(t,0)=f(t,\alpha(t))\equiv 0$, $f_u(t,u)<0$ for any $t\in [0,T]$ and $u\in [\alpha_0, \alpha^0]$, $f(t,u)<0$ for $u>\alpha(t)$, and \begin{equation}\label{main-assump1} \int_u^{\alpha_0}\min_{t\in[0,T]}f(t,s)ds>0 \quad\text{for all } u\in [0, \alpha_0). \end{equation} \end{itemize} In the special case where $f(t,u)=u(a-bu)(a, b>0)$, the problem \eqref{main-eq} was studied in \cite{DuLi}. Such a problem can be regarded as a model describing the spreading of a new or invasive species over a one-dimensional habitat, where $u(t,x)$ represents the density of the species at location $x$ and time $t$, and its spreading fronts are represented by the free boundaries $x=g(t)$ and $x=h(t)$. The Stefan conditions $g'(t)=-\mu u_x(t,g(t))$ and $h'(t)=-\mu u_x(t,h(t))$ are interpreted as saying that the spreading fronts expand at a speed proportional to the population gradient at the front, a deduction of these conditions from ecological considerations can be found in \cite{BuDuKr}. Among others, Du and Lin \cite{DuLi} proved a spreading-vanishing dichotomy result for the asymptotic behavior of the solutions, namely, there is a barrier $R^*>0$ such that \begin{itemize} \item[(i)] Spreading: the spreading fronts break the barrier at some finite time, and then the free boundaries go to infinity (i.e., $-g(t), h(t)\to\infty$ as $t\to\infty$), and the population successfully establishes itself in the new environment (i.e., $u(t,x)\to a/b$ as $t\to\infty$). \item[(ii)] Vanishing: the fronts never break the barrier (i.e., $h(t)-g(t)\alpha_0, \quad \int_u^{\alpha_0}\tilde{\rho} (s)ds>0 \text{ for } 0\le u<\alpha_0. $$ We take a $C^1$ function $\rho(u)$ such that it is slightly smaller than $\tilde{\rho}$, $\rho'(\alpha_0) <0$, and that, for given small $\varepsilon>0$ and $\alpha_\varepsilon := \alpha_0 - \varepsilon$, \begin{equation}\label{exis-of-homoge-1} \rho (\alpha_\varepsilon)=0, \; \rho(u)<0 \text{ for } u>\alpha_\varepsilon, \quad \int_u^{\alpha_\varepsilon}\rho (s)ds>0 \text{ for } 0\le u<\alpha_\varepsilon. \end{equation} Denote $$ \theta:=\max\{u<\alpha_\varepsilon:\rho(u)=0\},\quad \bar{\theta}:=\inf\{ u>\theta:\int_0^u \rho(s)ds>0\}. $$ Then $\theta\in [0,\alpha_\varepsilon)$ and $\bar{\theta} \in [\theta, \alpha_\varepsilon)$. We will show in Lemma \ref{sta-solu} that, for each $\beta \in (\bar{\theta}, \alpha_\varepsilon)$, the problem \begin{equation}\label{comp supp sol} v''+ \rho(v)=0, \quad v(0)=\beta,\quad v'(0)=0 \end{equation} has a unique solution $V(x;\beta)$, positive in $(-L,L)$ for some $L>0$ and $V(\pm L;\beta)=0$. Clearly, each one of such functions is a subsolution of the original equation \eqref{main-eq}$_1$. \item[(3)] \textbf{Periodic rightward traveling semi-wave}. Consider the problem \begin{equation} \label{r-semi-wave} \begin{gathered} U_t=U_{zz}-rU_z+f(t,U), \quad t\in[0,T],z>0,\\ U(t,0)=0,U(t,\infty)=P(t),\quad t\in[0,T],\\ U(0,z)=U(T,z),U_z(t,z)>0,\quad t\in[0,T],z>0,\\ r(t)=\mu U_z(t,0),\quad t\in[0,T]. \end{gathered} \end{equation} We will show in Proposition \ref{semi-wave-exist} that this problem has a solution pair $(r,U)$ with $r=r(t)\in\mathcal{P}_+$, where \begin{gather*} \mathcal{P}:=\{p\in C^{\gamma/2}([0,T]):p(0)=p(T)\}, \\ \mathcal{P}_+:=\{p\in\mathcal{P}:p(t)>0 \text{ for all } t\in[0,T]\}. \end{gather*} With $R(t):=\int_0^t r(s)ds$, the function $u(t,x)=U(t,R(t)-x;-r)$ satisfies \eqref{main-eq}$_1$, $u(t,R(t))=0$ and $R'(t)=-\mu u_x(t,R(t))$. We call $u=U(t,R(t)-x;-r)$ a \emph{periodic rightward traveling semi-wave} since it is only defined in $x\le R(t)$ and $U(t,z;-r)$ is periodic in $t$. \end{itemize} Throughout this article we choose the initial data $u_0$ from the set \begin{equation}\label{initial-value} \begin{split} \mathcal{X}(h_0)=\big\{&\phi\in C^2([-h_0,h_0]): \phi(-h_0)=\phi(h_0)=0,\phi'(-h_0)>0,\\ &\phi'(h_0)<0,\phi(x)>0 \text{ in } (-h_0,h_0)\big\}. \end{split} \end{equation} By a similar argument as in \cite{DuLou}, one can show that, for any $h_0>0$ and any initial data $u_0$, the problem \eqref{main-eq} has a time-global solution $(u(t,x),g(t),h(t))$, with $u\in C^{1+\gamma/2,2+\gamma}((0,\infty)\times[g(t),h(t)])$ and $g,h\in C^{1+\gamma/2}(0,\infty)$. Moreover, it follows from the maximum principal that, when $t>0$, the solution $u$ is positive in $(g(t),h(t))$, with $u_x(t,g(t))>0$ and $u_x(t,h(t))<0$. Thus $g'(t)<00$. Denote $$ g_\infty:=\lim_{t\to\infty}g(t), \quad h_\infty:=\lim_{t\to\infty}h(t), \quad I_\infty:=(g_\infty,h_\infty) $$ There are some possible situations on the asymptotic behavior of the solutions to \eqref{main-eq}. Spreading phenomenon is the most interesting one among them. Our first main result provides some sufficient conditions for spreading. \begin{theorem} \label{main-thm0} Assume {\rm (H1)}. If $u_0\in\mathcal{X}(h_0)$ satisfies $u_0 \geq V(x;\beta)$, where $V$ is the unique solution of the problem \eqref{comp supp sol} for some $\beta \in(\bar{\theta},\alpha_\varepsilon)$, then spreading happens in the sense that $ h_\infty=-g_\infty=\infty$, and \begin{equation} \label{spreading-happening} \lim_{t\to\infty}[u(t,\cdot)-P(t)]=0 \quad \text{locally uniformly in } \mathbb{R}. \end{equation} \end{theorem} Furthermore, when spreading happens, we will show that the right front of $u \approx U(t,R(t)-x)$ and the left front of $u \approx U(t,x+R(t))$. To construct precise sub- and supersolutions in our approach, we need the exponential stability of $P(t)$. For this purpose we have an additional condition: \begin{itemize} \item[(H2)] the function $f$ satisfies \begin{equation} \label{main-assump2} \sigma(t):=f_u(t,P(t))-\frac{f(t,P(t))}{P(t)}<0 \quad \text{for } t\in[0,T]. \end{equation} \end{itemize} \begin{theorem} \label{main-thm} Assume {\rm (H1), (H2)}. When spreading happens, there exists $H_1,G_1\in\mathbb{R}$ such that \begin{gather} \label{r-free-boun-bounded} \lim_{t\to\infty}[h(t)-R(t)]=H_1, \quad \lim_{t\to\infty}[h'(t)-r(t)]=0, \\ \label{l-free-boun-bounded} \lim_{t\to\infty}[g(t)+R(t)]=G_1, \quad \lim_{t\to\infty}[g'(t)+r(t)]=0, \\ \label{r-semi-wave-conver} \lim_{t\to\infty}\|u(t,\cdot)-U(t,R(t)+H_1-\cdot)\|_{L^\infty([0,h(t)])}=0, \\ \label{l-semi-wave-conver} \lim_{t\to\infty}\|u(t,\cdot)-U(t,\cdot+R(t)-G_1)\|_{L^\infty([g(t),0])}=0, \end{gather} where $R(t)=\int_0^t r(s)ds$. Here we extend $U(t,z)$ to be zero for $z<0$. \end{theorem} This article is organized as follows. In Section \ref{sec:spreading}, we present the lower and upper estimates for the solution to \eqref{main-eq} and prove Theorem \ref{main-thm0}. In Section 3, we construct a time-periodic traveling semi-wave and use it to characterize the profile of the spreading solutions, and prove Theorem \ref{main-thm}. \section{Spreading happening} \label{sec:spreading} In this section, we give sufficient conditions to ensure the spreading phenomena happens. We first construct some subsolutions of \eqref{main-eq}$_1$ which will be used for comparison, then we present the lower and upper estimates for $u$ and prove Theorem \ref{main-thm0}. Throughout this section, we assume (H1) and use the notation $\rho, \alpha_\varepsilon, \theta, \bar{\theta}$ etc.\ as in Section 1. \subsection{Subsolutions}\label{subsec:lower sol} In this subsection, we construct the subsolutions to \eqref{main-eq}$_1$, which are solutions to $v''+ \rho(v)=0$ with compact supports. \begin{lemma}\label{sta-solu} For any $\beta\in(\bar{\theta},\alpha_\varepsilon)$, the unique solution $V(x;\beta)$ of \eqref{comp supp sol} exists in the interval $[-L,L]$ for some $L=L(\beta)>0$, and \begin{equation}\label{property of V} V(\pm L;\beta) =0,\quad V(x;\beta) = V(-x;\beta),\quad V'(x;\beta)<0 \quad \text{for } 00$ such that the problem \eqref{comp supp sol} has a solution $V(x;\beta)\in C^2 ([-L(\beta),L(\beta)])$ with $V( \pm L(\beta);\beta)= 0$. Obviously, $V=V(x;\beta)$ satisfies the properties \eqref{property of V}. \end{proof} Collecting the solutions of \eqref{comp supp sol} for different $\beta$ we obtain a set $$ \mathcal{S}=\{v : v = V(x;\beta) \text{ for some } \beta\in (\bar{\theta}, \alpha_\varepsilon)\} . $$ The previous lemma indicates that this set is not empty. Moreover, from the above phase plane analysis, it is easily seen that $L(\beta)$ is continuous in $\beta \in (\bar{\theta}, \alpha_\varepsilon)$ and, as $\beta \to \alpha_\varepsilon$, $L(\beta) \to \infty$ and $V(x;\beta)\to \alpha_\varepsilon$ in $L^\infty_{\rm loc}(\mathbb{R})$ topology. \subsection{Lower estimate} Since $\alpha_0 > \alpha_\varepsilon >\bar{\theta}$, we have $\delta := (\alpha_0 - \bar{\theta})/3 >0$. When we take $\varepsilon>0$ small in $\alpha_\varepsilon$ we have $\alpha_0 -\delta < \alpha_\varepsilon <\alpha_0$. Now we show that spreading happens for the solution $(u,g,h)$ of \eqref{main-eq} in a weak sense. \begin{lemma}\label{lem:weak spreading} Let $(u,g,h)$ be the solution triple of problem \eqref{main-eq} with initial data $u_0$ as in Theorem \ref{main-thm0}. Then $h_\infty = -g_\infty =\infty$, and for any integer $n$, there exists $\tau(n)>0$ such that \begin{equation}\label{u>a0-2delta} u(t,x)\geq \alpha_0 -2\delta, \quad x\in [-n, n],\; t\geq \tau(n). \end{equation} \end{lemma} \begin{proof} Let $u_0$ be the initial data in Theorem \ref{main-thm0}. Consider the auxiliary problem \begin{equation}\label{FBP rho} \begin{gathered} w_t=w_{xx}+\rho (w), \quad \tilde{g}(t)0,\\ w(t, \tilde{g}(t) )=0, \quad \tilde{g}'(t)= -\mu w(t,\tilde{g}(t)), \; t>0,\\ w(t, \tilde{h}(t) )=0, \quad \tilde{h}'(t)= -\mu w(t,\tilde{h}(t)), \; t>0,\\ -\tilde{g}(0)=\tilde{h}(0)=h_0, \quad w(0,x)= u_0(x), \quad -h_0\le x \le h_0. \end{gathered} \end{equation} By \cite[Theorem 1.1]{DuLou}, either $\tilde{h}(t)-\tilde{g}(t)$ remains bounded and $w(t,\cdot)\to 0$ as $t\to \infty$, or $\tilde{h}(t), -\tilde{g}(t)\to \infty $ and $w(t,\cdot)$ converges to a stationary solution $w_\infty (x)$ as $t\to \infty$. In particular, if $u_0 (x)\geq V (x;\beta)$ as in Theorem \ref{main-thm0}, we have $w_\infty (x) \geq V(x;\beta)$ by the comparison principle, and so $\tilde{h}(t), -\tilde{g}(t)\to \infty$. Therefore, $w_\infty (x)$ is a solution of $v''+\rho(v)=0$, positive in $\mathbb{R}$ and larger than $V(x;\beta)$, which is nothing but $\alpha_\varepsilon$. This implies that, for any integer $n$, there exists $\tau (n) >0$ such that $$ w(t,x)\geq \alpha_\varepsilon - \delta ,\quad x\in [-n,n],\ t\geq \tau (n). $$ Since $\rho(u) \leq f(t,u)$ by the definition of $\rho$, we see that $(w,\tilde{g},\tilde{h})$ is a subsolution of \eqref{main-eq}, and so \begin{gather*} h(t) \geq \tilde{h}(t)\to \infty,\quad g(t)\leq \tilde{g}(t)\to -\infty \quad \text{as } t\to \infty, \\ u(t,x)\geq w(t,x) \geq \alpha_\varepsilon -\delta > \alpha_0 -2\delta, \quad x\in [-n, n],\; t\geq \tau (n). \end{gather*} This completes the proof. \end{proof} In the rest of this subsection we will use \eqref{u>a0-2delta} to give the lower estimate $P(t)-\epsilon$ for $u$. Define \begin{gather} k_1 (t,\eta)= \frac{f(t,\alpha_0)}{2\delta } [ \eta- (\alpha_0 -2\delta) ], \quad t\in [0,T],\ \eta\in \mathbb{R}, \nonumber \\ \label{def-k} k(t,\eta) = \min\{ k_1 (t,\eta),\ f(t,\eta)\},\quad t\in [0,T],\; \eta\geq \alpha_0 -2\delta. \end{gather} For any large integer $n>0$, let $\tau(n) $ be the time in \eqref{u>a0-2delta}, then there exists a large integer $k_n $ such that $k_n T > \tau (n)$. Consider the problem \begin{equation} \label{mono-sub-eq} \begin{gathered} \eta_t=\eta_{xx}+k(t,\eta), \quad -n0,\\ \eta(t,\pm n)= \alpha_0 -2\delta, \quad t>0, \\ \eta(0,x)=u(k_n T,x), \quad -n\le x \le n. \end{gathered} \end{equation} By \cite[Theorem 1]{BrPoSa}, the solution $\eta(t,x;n)$ of \eqref{mono-sub-eq} converges as $t\to \infty$ to a time-periodic solution $\eta^{\rm per} (t,x;n)$ of \eqref{mono-sub-eq}. Note that $k(t,\eta)$ is a Fisher-KPP type nonlinearity (above $\alpha_0 -2\delta$), by \cite{He,Na}, $\eta^{\rm per}(t,x;n)\to P(t)$ as $n\to \infty$ in $L^\infty_{\rm loc}(\mathbb{R})$ topology. \begin{lemma}\label{lem:lower bound} Under the assumption of Theorem \ref{main-thm0}, for any $\epsilon>0$ and any $M>0$, there exists $\tau(M,\epsilon)>0$ such that \begin{equation}\label{u>P-epsilon} u(t,x) \geq P(t)- \epsilon ,\quad x\in [-M, M],\ t\geq \tau(M,\epsilon). \end{equation} \end{lemma} \begin{proof} By $\eta^{\rm per}(t,x;n)\to P(t)$ as $n\to \infty$, there exists an integer $n_0 >M $ depending on $M$ and $\epsilon$ such that \begin{equation}\label{est 1} \eta^{\rm per}(t,x;n_0) > P(t) - \frac{\epsilon}{2},\quad x\in [-M, M],\; t\in [0,T]. \end{equation} For this fixed $n_0$, we see that the solution $\eta(t,x;n_0)$ of \eqref{mono-sub-eq} (with $n=n_0$) converges as $t\to \infty$ to $\eta^{\rm per}(t,x;n_0)$. Thus, there exists a integer $n_1$ such that for any integer $m\geq n_1$ we have \begin{equation}\label{est 2} \eta(mT +t, x;n_0) \geq \eta^{\rm per}(t,x;n_0) -\frac{\epsilon}{2},\quad x\in [-n_0, n_0],\ t\in [0,T]. \end{equation} Finally, using \eqref{u>a0-2delta} and the comparison principle to compare $u(k_{n_0} T+t, x)$ with the solution $\eta(t,x;n_0)$ of \eqref{mono-sub-eq} we have \begin{equation}\label{est 3} u(k_{n_0} T + t, x)\geq \eta(t,x;n_0),\quad x\in [-n_0, n_0],\ t>0. \end{equation} Combining the inequalities in \eqref{est 1}, \eqref{est 2} and \eqref{est 3} we obtain $$ u(k_{n_0} T +mT + t, x) \geq P(t) - \epsilon,\quad x\in [-M,M],\ t\in [0,T],\ m\geq n_1. $$ Choosing $\tau(M,\epsilon) = k_{n_0} T + n_1 T$ we obtain \eqref{u>P-epsilon}. \end{proof} \begin{proof}[Proof of Theorem \ref{main-thm0}] Consider the initial value problem, of ODE, \begin{equation} \begin{gathered} \zeta_t= f(t,\zeta),\quad t>0,\\ \zeta(0)= \alpha^0 + \| u_0\|_\infty . \end{gathered} \end{equation} It is known that $\zeta(t)$ decreases for small $t$ and then converges to $P(t)$ as $t\to \infty$. Hence for any small $\epsilon >0$, there exists $\tau_1 >0$ such that $\zeta (t)\le P(t)+\epsilon$ when $t\geq \tau_1$. By comparison we have $$ u(t,x)\le \zeta (t) \leq P(t) +\epsilon, \quad x\in [g(t), h(t)],\ t\geq \tau_1. $$ Combining with \eqref{u>P-epsilon} we prove \eqref{spreading-happening}. This conclusion and Lemma \ref{lem:weak spreading} complete the proof. \end{proof} \section{Using semi-wave to characterize the spreading phenomena} In this section we first construct a time-periodic traveling semi-wave $U$ propagating rightward with speed $r(t)$, and then prove the boundedness of $|h(t) - \int_0^t r(s)ds|$ and $|g(t)+ \int_0^t r(s)ds|$ by using the method of lower and upper solutions as in \cite{DuMaZh}. At last, we characterize the fronts of spreading solutions by the semi-wave and prove Theorem \ref{main-thm}. \subsection{Time-periodic traveling semi-wave} In this subsection we construct a traveling semi-wave which is periodic in time and is used to characterize the spreading solutions near the boundaries. Our approach is similar as that in \cite{SuLoZh}. For readers' convenience, we present the details. Let $\mathcal{P}$ be the set of periodic functions defined as in section 1, $P(t)$ be the largest periodic solution to $u_t = f(t,u)$. First, we consider the following problem \begin{equation} \label{perio-solu-bounded} \begin{gathered} v_t=v_{zz}+k(t)v_z+f(t,v), \quad t\in[0,T],\; z\in(0,l),\\ v(t,0)=0, v(t,l)=P^0:=\max_{0\le t\le T}P(t), \quad t\in[0,T],\\ v(0,z)=v(T,z),\quad z\in[0,l]. \end{gathered} \end{equation} \begin{lemma} For any $k\in\mathcal{P}$ and any $l>0$, the problem \eqref{perio-solu-bounded} has a maximal solution $v=U_1(t,z;k,l)$, which is strictly increasing in both $z\in[0,l]$ and $k\in\mathcal{P}$, strictly decreasing in $l>0$. \end{lemma} \begin{proof} Consider the equation and the boundary condition in \eqref{perio-solu-bounded} with initial data $v(0,z):=P^0\cdot \chi_{[0,l]}(z)$, which is the characteristic function on the interval $[0,l]$. This initial boundary value problem has a unique solution $v(t,z;k,l)$. Using the maximum principle we see that $v(t,z;k,l)$ is strictly increasing in $z\in[0,l]$ and $k\in\mathcal{P}$, strictly decreasing in $l>0$, and $v(t,z;k,l)\le P^0$. Using the zero number argument in a similar way as in the proof of \cite[Theorem 1]{BrPoSa} one can show that $||v(t,\cdot;k,l)-U_1(t,\cdot;k,l)||_{C^2([0,l])}\to0$ as $t\to\infty$, where $U_1(t,z;k,l)\in C^{1+\gamma/2,2+\gamma}([0,T]\times[0,l])$ is a time-periodic solution of \eqref{perio-solu-bounded}. By the maximum principle again, we see that $U_1$ has the same monotonic properties as $v$ in $z$, $k$ and $l$. \end{proof} Next, we consider the problem on the half line, \begin{equation} \label{perio-solu-unbounded} \begin{gathered} v_t=v_{zz}+k(t)v_z+f(t,v), \quad t\in[0,T],z>0,\\ v(t,0)=0, \quad t\in[0,T],\\ v(0,z)=v(T,z),\quad z\ge0. \end{gathered} \end{equation} \begin{lemma} \label{bound-nonnega-perio-solu} For each $k\in\mathcal{P}$, problem \eqref{perio-solu-unbounded} has a maximal bounded and nonnegative solution $U(t,z;k)$ with $U_z(t,z;k)\ge0$ in $[0,T]\times[0,\infty)$. $U_z(t,0;k)$ is continuous in $k$ in the sense that, for $\{k_1,k_2,\dots\}\subset\mathcal{P}$, $U_z(t,0;k_n)\to U_z(t,0;k)$ in $C^{\gamma/2}([0,T])$ if $k_n\to k$ in $C^{\gamma/2}([0,T])$. Assume further that $k\ge0$. Then $U_z(t,z;k)>0$ in $[0,T]\times[0,\infty)$, $U(t,z;k)-P(t)\to 0$ as $z\to\infty$. $U_z(t,0;k)$ has a positive lower bound $\delta$ (independent of $t$), and it is strictly increasing in $k$: $U_z(t,0;k_1)\frac{L(\beta)}{2}. $$ Hence, $U(t,z;k)\ge V(z-L(\beta);\beta)$. It means that $U_z(t,0;k)\ge \delta:=V'(-L(\beta);\beta)>0$. Using the strong maximum principle to $U_z$ we conclude that $U_z(t,z;k)>0$ in $[0,T)\times[0,\infty)$. Thus $P_1:=\lim_{z\to\infty}U(t,z;k)$ exists. In a similar way as in the proof of \cite[Proposition 2.1]{DuGuPe} one can show that $P_1(t)$ is nothing but the maximal positive periodic solution $P(t)$ of $u_t=f(t,u)$. Since $U(t,z;k)$ is non-decreasing in $k$ we have $U_z(t,0;k_1)\le U_z(t,0;k_2)$ when $k_1\le k_2$. The strict inequality $U_z(t,0;k_1)0$. When $r=A[0]$ we have $A[-A[0]]=\mu U_z(t,0;-A[0])\ge0$ and $A[-A[0]]\le A[0]$. Set $\mathcal{R}:=[0,A[0]]$, then as in the proof of \cite[Theorem 2.4]{DuGuPe} one can show that the mapping $A[-\cdot]$ maps $\mathcal{R}$ continuously into a precompact set in $\mathcal{R}$. Using the Schauder fixed point theorem we see that there exists $r(t)\in\mathcal{R}$ such that $r(t)=A[-r](t)$. Clearly, $r(t)\ge0$. Obviously, $r(t)\equiv0$ is impossible since $A[0]>0$. If $r(t)\ge,\neq0$, the strong maximum principle and Hopf Lemma tells us that $U_z(t,0;-r)>0$, so it contradicts to $r(t)=A[-r](t)$. This yields $r(t)\in P_+$. Finally, a direct calculation shows that the function $u=U(t,R(t)-x;-r)$ with $R(t):=\int_0^t r(s)ds$ solves the equation \eqref{main-eq}$_1$ in $\mathbb{R}\times (-\infty,R(t))$. \end{proof} \subsection{Boundedness for $|h(t)-R(t)|$ and $|g(t)+R(t)|$} Let $U(t,R(t)-x;-r)$ be the rightward periodic traveling semi-wave with speed $r(t)$, where $R(t):=\int_0^t r(s)ds$. We show that $|h(t)-R(t)|$ and $|g(t)+R(t)|$ are both bounded for all $t\ge0$. \begin{lemma} Assume that {\rm (H1), (H2)}. There exists $C>0$ such that \begin{equation} \label{h-R} |h(t)-R(t)|,\ |g(t)+R(t)|\leq C \quad \text{for all } t\geq0. \end{equation} \end{lemma} We will show the boundedness of $|h(t)-R(t)|$ only, since the situation for $|g(t)+R(t)|$ can be proved similarly. For convenience, write $v(t,x):=\frac{u(t,x)}{P(t)}$, $\nu(t):=\mu P(t)$ to normalize the problem \eqref{main-eq} into \begin{equation} \label{u-norm} \begin{gathered} v_t=v_{xx}+F(t,v),\quad t>0,\; g(t)0,\\ v(t,g(t))=0, g'(t)=-\nu(t) v_x(t,g(t)), \quad t>0,\\ v(0,x)=u_0(x)/P(0),\quad -h_0\leq x\leq h_0, \end{gathered} \end{equation} where $F(t,v)=\frac{1}{P(t)}[f(t,P(t)v)-f(t,P(t))v]$ satisfying $F(t,v)\in C^{\gamma/2,1}_{\rm loc}([0,T]\times\mathbb{R})$ for some $\gamma\in(0,1)$, $T$-periodic in $t$, $F(t,0)\equiv F(t,1)\equiv 0$. Then by assumption (H2) and the fact that $P(t)>0$, there exists $\delta>0$ such that $F_v(t,1)=\frac{P(t)f_u(t,P(t))-f(t,P(t))} {P(t)}<-2\delta$, so we can find some $\epsilon>0$ such that \begin{equation} \label{F_v(t,1)} F_v(t,v)\leq -\delta \text{ for } t\in[0,T],\ v\in[1-\epsilon,1+\epsilon]. \end{equation} Consider the solution $\xi (t)$ of ODE $\xi_t=F(t,\xi)$ with initial value $\xi (0)=M/P(0)+1$, where $M=\|u_0\|_{L^\infty([-h_0,h_0])}+1$. Clearly, $\xi (t)$ decreases to $1$ as $t \to \infty$ by the uniqueness of $P(t)$. Hence for $\epsilon>0$ given in \eqref{F_v(t,1)}, one can choose a large integer $m$ such that $1<\xi(t)<1+\epsilon$ and $\xi_t=F(t,\xi)\leq \delta (1-\xi)$ for $t\geq mT$. So $\xi(t)\leq 1+\epsilon e^{\delta (mT - t)}$ for $t\geq mT$. Then by the comparison principle, we have $$ v(t,x)\leq \xi(t) \leq 1+\epsilon e^{\delta (mT - t)} \quad \text{for } g(t)\leq x \leq h(t),\ t\geq mT. $$ Now we also normalize the periodic rightward semi-wave $U(t,z;-r)$ by setting $V(t,z):=\frac{U(t,z;-r)}{P(t)}$, then $V(t,z)$ satisfies \begin{equation} \begin{gathered} V_t=V_{zz}-r(t)V_z+F(t,V), \quad t\in[0,T],z>0,\\ V(t,0)=0,V(t,\infty)=1, \quad t\in[0,T],\\ V(0,z)=V(T,z),V_z(t,z)>0, \quad t\in[0,T],z>0,\\ r(t)=\nu(t) V_z(t,0), \quad t\in[0,T]. \end{gathered} \end{equation} Then we can find an integer $m_1>m$ and a constant $X>0$ large enough such that \begin{equation} (1+M_1e^{-\delta T_1})V(t,z)\geq 1+\epsilon e^{\delta (mT - T_1)} \quad \text{for all } t\in [0,T],\; z\geq X. \end{equation} where $M_1=2\epsilon e^{\delta mT}$, $T_1=m_1T$. We construct a supersolution $(v^+,g,h^+)$ to \eqref{u-norm} as follows: let \begin{gather*} h^+(t):=\int_{T_1}^{t}r(s)ds+h(T_1)+KM_1(e^{-\delta T_1}-e^{-\delta t})+X, \\ v^+(t,x):=(1+M_1e^{-\delta t})V(t,h^+(t)-x). \end{gather*} where $K$ is a positive constant that can be chosen sufficiently large. By direct computations, one can easily check that \begin{gather*} v^+_t-v^+_{xx}\geq F(t,v^+) \quad \text{for } t\geq T_1,\; g(t)-\gamma (t)v^+(t,x) \quad \text{for } t\geq T_1,\ x=h^+(t), \\ h(T_1)\leq h^+(T_1),v(T_1,x)\leq v^+(T_1,x) \quad \text{for } x \in [g(T_1),h(T_1)]. \end{gather*} \begin{proposition} For sufficiently large $K>0$, the function $h(t)$ satisfies \begin{equation} h(t)0$ given above, there exists some $c>0$ and $K_1>0$ such that \begin{equation} \label{u(t,x)-P(t)} \|u(t,\cdot)-P(t)\|_{L^{\infty}([-ct,ct])}\leq K_1e^{-\delta t} \quad \text{for } t\gg 1. \end{equation} \end{proposition} \begin{proof} For $c>0$ small enough and some suitable $\epsilon_1$ to be determined below, by simple phase plane analysis, the problems \begin{equation} \begin{gathered} q_{zz}\pm c q_z + \rho(q)=0, \quad z\in [-L,L],\\ q(\pm L)=0,\quad \max q(z) = \alpha_{\varepsilon} -\epsilon_1, \end{gathered} \end{equation} have the solutions $q_{\pm}(z)$ respectively. Therefore, $w(t,x)=q_+(x-ct)$ and $w(t,x)=q_-(x+ct)$ are two compactly supported \emph{traveling wave} solutions to the problems \begin{equation} \begin{gathered} w_t= w_{xx}+ \rho(w), \quad x\in [\pm ct-L,\pm ct+L],\; t\in \mathbb{R},\\ w(t, \pm ct+ L)=0 ,\quad w(t, \pm ct- L)=0, \quad t\in \mathbb{R},\\ \max w(t,x)= \alpha_{\varepsilon} -\epsilon_1, \quad t\in \mathbb{R}. \end{gathered} \end{equation} Since spreading happens for the solution $(u,g,h)$, there exists a large integer $m_0$ such that $$ u(mT,x) > \alpha_{\varepsilon} -\epsilon_1 \quad \text{for all } m\geq m_0,\; x\in [-cT-L,cT+L]. $$ Thus, we have $$ u(mT,x) > q_{\pm}(x+x_0) \quad \text{for all } m\geq m_0,\; x\in [-cT-L,cT+L],\ x_0\in[-cT,cT]. $$ Then by the comparison principle, \[ u(mT+t, x) \geq q_+(x+x_0-ct), \; q_-(x+x_0+ct) \] for all $ t>0$ and $x\in[-L-cT \pm ct,L+cT\pm ct]$. Using this inequality it is easily to show that $$ u(mT+t, x) \geq \alpha_{\varepsilon} -\epsilon_1\quad \text{for } t>0,\; x\in [-ct -L, ct+L]. $$ Since $\epsilon$ and $\epsilon_1$ can be chosen sufficiently small, using the same argument for normalized function $v$ as in the proof of \cite[Lemma 6.5]{DuLou}, one can check that $$ |v(t,x)-1|\leq k_1e^{-\delta t} \quad \text{for } x\in [-ct,ct],\; t\geq T_2 := m_2 T, $$ where $m_2 > m_0$ is an integer and $k_1>0$ is a constant sufficiently large. This reduces to \eqref{u(t,x)-P(t)}. Let $c,K_1$ and $\delta$ be the constant as before, we define \begin{gather*} g^-(t):=0, \quad h^-(t):=\int_{T_2}^{t}r(s)ds+h(T_1)-K_2K_1(e^{-\delta T_2}-e^{-\delta t})+cT_2, \\ v^-(t,x):=(1-K_1e^{-\delta t})V(t,h^-(t)-x). \end{gather*} Then for a suitable constant $K_2>0$, by the similar argument as in the construction of supersolution, one can show that $(v^-,g^-,h^-)$ is a subsolution. Hence $$ h(t)\geq h^-(t)-\max_{t\in [0,T_2]}|h(t)-h^-(t)|\geq R(t)-H_l \quad \text{for all } t\geq 0, $$ where $H_l:=\max_{t\in [0,T_2]}|h(t)-h^-(t)|+cT_2+K_2K_1$. Then we obtain \eqref{h-R}. \end{proof} \begin{proof}[Proof of Theorem \ref{main-thm}] We only prove \eqref{r-free-boun-bounded} and \eqref{r-semi-wave-conver}, since the proof for \eqref{l-free-boun-bounded} and \eqref{l-semi-wave-conver} are similar. By using changing the coordinate $y:=x-R(t)$, we set \begin{gather*} h_1(t):=h(t)-R(t),\quad g_1(t):=g(t)-R(t) \quad \text{for } t\geq 0, \\ u_1(t,y):=u(t,y+R(t)) \quad \text{for } t>0, \; y\in [g_1(t),h_1(t)]. \end{gather*} Also for any constant $y_0\in R$, we define $V_1(t,y):=U(t,y_0-y;-r)$, which is a rightward periodic traveling semi-wave with speed $r(t)$. Consider the zero number of function $\eta_1(t,y):=u_1(t,y)-V_1(t,y)$ in the moving area $J(t):=[g(t),\min(y_0,h(t))]$, and denote by $Z_{J(t)}[\eta_1]$ the zero number of $\eta_1(t,\cdot)$ in the interval $J(t)$. Then the zero number argument yields that $Z_{J(t)}[\eta_1]$ is finite and decreases strictly when $h_1(t)$ gets across $y_0$. So $h_1(t)-y_0$ changes sign at most finite times, namely, $h_1(t)>y_0$ or $h_1(t)0$, $$ \|h'(t)\|_{C^{\frac{\gamma}{2}}([\tau,\tau+1])}\leq C, $$ where $C>0$ is independent of $\tau$. Combining with the convergence of $h_1(t)$ we have $h_1'(t)\to 0$ as $t\to \infty$, that is, $\lim_{t\to \infty}[h'(t)-r(t)]=0$. Next, we prove \eqref{r-semi-wave-conver}. We use the variable substitution $z:=x-h(t)$ to set \begin{gather*} g_2(t):=g(t)-h(t) \quad \text{for } t\geq 0, \\ u_2(t,z):=u(t,z+h(t)) \quad \text{for } t>0, \; z\in [g_2(t),0]. \end{gather*} Then the rightward free boundary of $u$ is fixed at $z=0$ and $(u_2,g_2)$ satisfies \begin{equation} \label{u_2g_2} \begin{gathered} u_{2t}=u_{2zz}+h'(t)u_{2z}+f(t,u_2), \quad t>0, \; g_2(t)0, \; z=g_2(t),\\ u_2(t,0)=0,h'(t)=-\mu u_{2z}(t,0), \quad t>0. \end{gathered} \end{equation} By $L^p$ theory and Soblev embedding theorem, for any constant $K>0$, there exists a sequence ${m_n}$ with $m_n\to \infty$ such that $$ \|u_2(m_nT+t,z)\|_{C^{1+\frac{\gamma}{2},2+\gamma}([-K,K]\times[-K,0])}\leq C, $$ where $C>0$ is a constant independent of $n$. By using Cantor's diagonal argument, there is a function $w(t,z)\in C^{1+\frac{\gamma}{2},2+\gamma}(R\times (-\infty,0])$ and a subsequence of ${m_n}$, denote again by ${m_n}$, such that $$ \lim_{n\to \infty}\|u_2(m_nT+t,z)-w(t,z)\|_{C^{1,2}_{\rm loc}(R\times (-\infty,0])}=0. $$ Replacing $t$ by $m_nT+t$ in \eqref{u_2g_2} and taking limit as $n\to \infty$, we obtain \begin{gather*} w_t=w_{zz}+r(t)w_z+f(t,w), \quad -\infty