\begin{array}{lll}
{\|u_t\|}_{C^{\beta}(\mathcal{K})} \le c_{8}[{\|u_t\|}_{L^1(\Omega)}+{\|g(u_t)\|}_{L^{\infty}(\mathcal{K}_{3t_0})}+{\|u_k\|}_{L^{\infty}(\mathcal{K}_{3t_0})}]
 \\[2mm]\phantom{-}\le
 c_{12}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}+\|g(\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|)]\|_{L^\infty( \mathcal{K}_{3t_0})}].
 \end{array}$$ We next claim that $\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}$, ${\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}$ are uniformly bounded. In fact, for $x\in\mathcal{K}$ and $y\in\partial\Omega_t$ with $t\in(0,t_0)$, we have that $|x-y|\ge 3t_0$. By (\[annex 01\]), it implies that $$\label{2.1.3}
\arraycolsep=1pt
\begin{array}{lll}
\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|](x)\le \int_{\partial\Omega_t}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t(y)|
 \\[3mm]\phantom{------} \le c_5t_0^{\alpha-N} {\|\nu_t\|}_{\mathfrak{M}^b(\bar\Omega)} = c_5t_0^{\alpha-N} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}
 \end{array}$$ and $$\label{2.1.4}
\arraycolsep=1pt
\begin{array}{lll}
{\|\mathbb{G}_{\alpha}[t^{-\alpha}\nu_t]\|}_{L^1(\Omega)}\le \int_{\Omega}\int_{\partial\Omega_t}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t(y)|dx
  \\[3mm]\phantom{------}=\int_{\partial\Omega_t}\int_{\Omega}\frac{c_5}{|x-y|^{N-\alpha}}dxd|\nu_t(y)|
\le c_{15} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)},
 \end{array}$$ which implies that $${\|u_t\|}_{C^{\beta}(\mathcal{K})}\le c_{15}{\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)},$$ where $c_{15}>0$ independent of $t$.

Moreover, if $g$ is $C^\beta$ locally in ${\mathbb{R}}$, similar to (\[2.0.10.0\]) it implies by (\[2.1.3\]) and (\[2.1.4\]) that $${\|u_t\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{16}{\|\nu\|}_{\mathfrak{M}^b(\bar
\Omega)},$$ where $c_{16}>0$ independent of $t$. We conclude by Theorem 2.2 in [@CFQ] that $u_t$ is a classical solution of $$\label{2.2.1}
\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+g(u)=0\quad & {\rm in} \quad\Omega\setminus\partial\Omega_t,\\[2mm]
 \phantom{   (-\Delta)^\alpha  + g(u)}
u=0\quad & {\rm in} \quad \Omega^c.
\end{array}$$ This ends the proof.$\Box$

\[pr 2.01\] Assume that $k>0$ and $\{g_n\}$ are a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ such that $g_n(0)=g(0)$ and (\[06-08-0\]). Then problem $$\label{10-08-0}
\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+ g_n(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad  &{\rm in}\quad\ \ \bar\Omega,\\[3mm]
 \phantom{(-\Delta)^\alpha   + g_n(u)}
 u=0\quad &{\rm in}\quad\ \ \bar\Omega^c
 \end{array}$$ admits a unique weak solution $u_{n}$ satisfying $$\label{10-08-1}
-k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_-}{\partial\vec{n}^\alpha}](x)\le u_n(x)\le k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_+}{\partial\vec{n}^\alpha}](x),\qquad x\in\Omega,$$ where $\nu_+,\nu_-$ are the positive and negative decomposition 