following follows from Lemmas A.2 and A.3 in [@xinwx1].

\[lexin3.4\]

(a) Let $a$, $b$ and $q$ be fixed positive constants. For any given $\varepsilon>0$ and $L>0$, there exists $$\displaystyle{l>\max\left\{L,\frac{\pi}{2}\sqrt{\frac{d}{a}}\right\}}$$ such that, if the continuous and non-negative function $U(t,x)$ satisfies $$\label{xin2.3}
    \left\{\begin{array}{ll}
    U_{t}-dU_{xx}\geq U(a-bU), &t>0,~~ 0<x<l,\vspace{7 pt}\\
    U_{x}(t,0)=0,U(t,l)\geq q,& t>0,~~ (q\geq0),
    \end{array}\right.$$ with $U(0,x)>0$ for all $x\in [0,l)$, then $$\liminf_{t\rightarrow+\infty}U(t,x)>\frac{a}{b}-
    \varepsilon~\text{uniformly~on}~
    [0,L].$$

(b) Let $a$, $b$ and $q$ be fixed positive constants. For any given $\varepsilon>0$ and $L>0$, there exists $\displaystyle{l> \max\left\{L,\frac{\pi}{2}\sqrt{\frac{d}{a}}\right\}}$ such that $$\limsup_{t\rightarrow+\infty}V(t,x)<\frac{a}{b}+\varepsilon
    ~\text{uniformly~on}~[0,L],$$ where $V(t,x)$ is a continuous and non-negative function satisfying $$\label{xin2.4}
    \left\{\begin{array}{ll}
    V_{t}-dV_{xx}\leq V(a-bV), &t>0, 0<x<l,\vspace{7 pt}\\
    V_{x}(t,0)=0,V(t,l)\leq q, &t>0,\;\; (q\geq0),
    \end{array}\right.$$ and $V(0,x)>0$ for all $x\in[0,l)$.

We are now in the position to present part of our main results.

The vanishing case {#the_vanishing}
------------------

We consider the vanishing case in this subsection.

\[th4.2\] Let $(u,v,h)$ be the solution of system with initial data . If $h_{\infty}<+\infty$, then $$\lim_{t\rightarrow +\infty} \|u(t,\cdot)\|_{C[0, h(t)]}=0~\text{ and }~\lim_{t\rightarrow +\infty}v(t,x)=\kappa_{2}$$ uniformly in any bounded subset of $[0,+\infty)$.

Theorem \[Thm\_existence\] yields that for $\theta \in (0,1)$, there is a constant $\hat{C}$ depending on $\theta$, $(u_{0}, v_{0})$, $h_{0}$ and $h_{\infty}$ such that $$\label{eqth4.2yong}
\|u\|_{C^{(1+\theta)/2,1+\theta}(G)}+\|v\|_{C^{(1+\theta)/2,1+\theta}(G)}
+\|h(t)\|_{C^{1+\theta/2}([0,\infty))}\leq \hat{C},$$ where $$G:=\{(t,x)\in[0,\infty)\times[0, h(t)]\}.$$ Suppose that $$\displaystyle{\limsup_{t \rightarrow 