sequence of nondecreasing functions $\{g_n\}\subset C^\beta({\mathbb{R}})$ such that $g_n(0)\ge0$, $|g_n(s)|\le |g(s)|$ for $s\in{\mathbb{R}}$ and $g_n\to g$ locally in ${\mathbb{R}}$ as $n\to\infty$.) Let $w_n$ be the classical solution of $$\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+g_n(u)=f_n\quad & {\rm in}\quad\Omega,\\[2mm]
 \phantom{   (-\Delta)^\alpha  + g_n(u)}
u=0\quad & {\rm in}\quad \Omega^c.
\end{array}$$ By the uniqueness of weak solution to (\[2.0.6\]), we obtain that, up to some subsequence, $$\label{12-08.2}
u_t=\lim_{n\to\infty} w_n\quad {\rm a.e.\ in}\quad \Omega.$$ We observe that $0\le w_n= \mathbb{G}_\alpha[f_n]-\mathbb{G}_\alpha[g(w_n)]\le \mathbb{G}_\alpha[f_n]$ and $\mathbb{G}_\alpha[f_n]$ converges to $\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]$ uniformly in any compact set of $\Omega\setminus \partial\Omega_t$ and in $L^1(\Omega)$, then there exists $c_{11}>0$ independent of $n$ and $t$ such that $${\|w_n\|}_{L^\infty(\mathcal{K}_{3t_0})}\le {c_{11}}{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^\infty(\mathcal{K}_{3t_0})},\quad  {\|w_n\|}_{L^1(\Omega)}\le {c_{11}}{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)}.$$ By [@CV3 Lemma 3.1], for $\beta\in(0,2\alpha)$, there exists $c_{12}>0$ independent of $n$ and $t$, such that $$\arraycolsep=1pt
\begin{array}{lll}
{\|w_n\|}_{C^{\beta}(\mathcal{K}_{2t_0})} \le c_8[{\|w_n\|}_{L^1(\Omega)}+{\|g(w_n)\|}_{L^{\infty}(\mathcal{K}_{3t_0})}+{\|w_n\|}_{L^{\infty}(\mathcal{K}_{3t_0})}]\\[3mm]
\phantom{} \le c_{12}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}+\|g(\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|])\|_{L^\infty( \mathcal{K}_{3t_0})}].
\end{array}$$ It follows by [@RS Corollary 2.4] that there exist $c_{13},c_{14}>0$ such that $$\label{2.0.10.0}
\arraycolsep=1pt
\begin{array}{lll}
{\|w_n\|}_{C^{2\alpha+\beta}(\mathcal{K})} \le {c_{13}}[{\|w_n\|}_{L^1(\Omega)}+{\|g(w_n)\|}_{C^{\beta}(\mathcal{K}_{2t_0})}
+{\|w_n\|}_{C^{\beta}(\mathcal{K}_{2t_0})}]\\[3mm]
\phantom{------\ } \le c_{14}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}\\[3mm]
\phantom{-------\ }+\|g\|_{C^\beta([0,\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^\infty( \mathcal{K}_{3t_0})}])} \|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{C^\beta( \mathcal{K}_{3t_0})}].
\end{array}$$ Therefore, together with (\[12-08.2\]) and the Arzela-Ascoli Theorem, it follows that $u_t\in C^{2\alpha+\epsilon}(\mathcal{K})$ for $\epsilon\in(0,\beta)$. Then $w_n\to u_t$ and $f_n\to 0$ uniformly in any compact subset of $\Omega\setminus\partial\Omega_t$ as $n\to\infty$. It infers by [@CV3 Lemma 3.1] that $$\arraycolsep=1pt
