Applying the comparison principle, we get $$\omega_{1}(t,x)\geq u(t,x),\text{ for }~t\in [0,T]~\text{ and  }~x\in [h(t)-M^{-1}, h(t)].$$ Since $\omega_{1}(t,h(t))=0=u(t,h(t))$, we then have $$u_{x}(t,h(t))\geq \omega_{1x}(t,h(t))=-2MM_{1}.$$ Consequently, $h'(t)=-\mu u_{x}(t,h(t))\leq \Lambda $ with $\Lambda:=2\mu MM_{1}$.

Bearing the above result in mind, we can show that the local solution obtained in Theorem \[Thm\_existence\] can indeed be extended to all $t > 0$.

\[Thm\_uniqueness\] System - admits a unique solution for $t\in [0, \infty)$.

Let $[0, T_{max})$ be the maximal time interval in which the unique solution exists. We will show that $T_{max}=\infty$. Suppose to the contrary that $T_{max}<\infty$. In view of Lemma \[le2.2\], there exists positive constants $M_{1}$, $M_{2}$ and $\Lambda$, independent of $T_{max}$, such that for $t\in [0,T_{max}]$, $$0<u(t,x)\leq M_{1},~0< v(t,x)\leq M_{2}~\text{ and }~0< h'(t)\leq \Lambda.$$ Fix $\delta \in (0,T_{max})$ and $K>T_{max}.$ Using the standard $L^{p}$ estimates together with the Sobolev embedding theorem and the Hölder estimates for parabolic equations (see Lunardi [@Lunardi] for eg.), we can find $M_{3}$ depending only on $\delta$, $K$, $M_{1}$ and $M_{2}$ such that $$\|u(t,\cdot)\|_{C^{1+\theta}[0, h(t))}\leq M_{3}\text{ and } \|v(t,\cdot)\|_{C^{1+\theta}[0,+\infty)}\leq M_{3} ~\text{ for all }~t\in [\delta, T_{max}),$$ where we used the convention that $u(t,x)=0$ for $x\geq h(t)$. By virtue of the proof of Theorem 2.1 in [@dusupandinf], there exists a 