vanishing case.

\[similarly with th4.2\] Let $(u,v,h)$ be the solution of system subject to initial conditions . If $h_{\infty}<+\infty$ and $b_{2}$ satisfies , where we replace $B(x)$ by $b_2(x)$, then $$\lim_{t\rightarrow +\infty} \|u(t,\cdot)\|_{C[0, h(t)]}=0~\text{ and }~\lim_{t\rightarrow +\infty}v(t,x)=\phi_{v*}$$ uniformly in any bounded subset of $[0,+\infty)$.

The proof is similar to that of Theorem \[th4.2\], above.

In order to obtain sharp criteria for spreading, we require stronger conditions on $b_{1}(x)$ and $\delta_{1}$. Namely, we assume that $$\label{assume}
 b_{1}(x)-\delta_{1}\phi_{v*}\text{ is positive somewhere in }[0,h_{0}].$$ Our assumption is not excessive in the sense that, when $b_{i}$ and $\delta_{i}~(i=1,2)$ are constant, we have $\phi_{v*}=\frac{b_{2}}{\delta_{2}}$. Consequently, $b_{1}-\delta_{1}\phi_{v*}$ is a positive constant over the interval $[0,+\infty)$.

\[the general case spreading\] Assume that $b_{1}(x)-\delta_{1}\phi_{v*}(x)$ satisfies and $b_{2}(x)$ satisfies (where $B$ is replaced accordingly). If $0<d_{1}<d_{1}^{*}$, then spreading occurs.

First, we consider the following equation: $$\left\{\begin{array}{ll}\label{aaa}
\displaystyle{\frac{\partial \overline{v}}{\partial t}(t,x)=d_{2}\overline{v}_{xx}+\overline{v}(b_{2}(x)-\delta_{2}\overline{v}),}&t>0, ~~ x>0,\vspace{7 pt}\\
\overline{v}_{x}(t,0)=0,&t>0,\vspace{7 pt}\\
\overline{v}(0,x)=v_{0}(x).
\end{array}\right.$$ Since $b_{2}(x)$ satisfies the hypotheses of Lemma \[the long time behavior\], all solutions of (\[aaa\]) with non-trivial non-negative initial values converge to $\phi_{v*}$ as $t\rightarrow \infty$.

It follows, from the comparison principle, that $v\leq\overline{v}$ for all $t>0$ and $x>0$. Since $\displaystyle{\lim_{t\rightarrow+\infty} \overline{v}(t,x)= \phi_{v*}}$ uniformly in any compact subset of $[0,\infty),$ then, for any 