where $M^{\frac{N}{N-\alpha}}(\Omega,\rho_{\partial\Omega}^\alpha dx)$ is Marcinkiewicz space with exponent $\frac{N}{N-\alpha}$. The definition and properties of Marcinkiewicz space see the references [@BBC; @CC; @CV1; @V].

In next lemma, the uniformly regularity plays an important role in our approximation of weak solution.

\[lm 1\] Assume that $u_{t}$ is a weak solution of (\[2.0.6\]) replacing $g_n$ by $g$, a continuous nondecreasing function satisfying $g(0)\ge 0$. Then for any compact subsets $\mathcal{K}\subset\Omega$, there exist $t_0>0$, $\beta>0$ small and $c_{10}>0$ independent of $t$ such that for $t\in(0,t_0]$, $$\label{2.0.8}
{\|u_{t}\|}_{C^{\beta}(\mathcal{K})}\le c_{10} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Moreover, if $g$ is $C^\beta$ locally in ${\mathbb{R}}$, then there exists $c_{11}>0$ independent of $t$ such that $$\label{2.0.9}
{\|u_{t}\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{11} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$

[**Proof.**]{} We observe from Proposition \[pr 1\] that $$\label{3--1}
| u_t|\le \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad  \Omega.$$ For compact set $\mathcal{K}$ in $\Omega$, there exists $t_0>0$ such that $$\mathcal{K}_{5t_0}\subset \Omega,$$ where $\mathcal{K}_r:=\{x\in{\mathbb{R}}^N:{\rm dist}(x,\mathcal{K})<r\}$ with $r>0$. Then $\mathcal{K}_{4t_0}\cap \partial\Omega_t=\O$ for any $t\in(0,t_0]$ and $${\|g(u_t)\|}_{L^\infty(\mathcal{K}_{3t_0})}\le {\|g(\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|])\|}_{L^\infty(\mathcal{K}_{3t_0})}.$$

Since $t^{-\alpha}\nu_t$ is a bounded Radon measure in $\Omega$, there exists a sequence $\{f_n\}\subset C^2_0(\Omega)$ such that $f_n$ converges to $t^{-\alpha}\nu_t$ in the distribution sense and for some $N_{t_0}>0$ such that for $n\ge N_{t_0}$, supp$(f_n)\cap \mathcal{K}_{3t_0}=\O$.

We may assume that $g$ is $C^\beta$ locally in ${\mathbb{R}}$. (In fact, we can choose a 