\epsilon}(0,x)
  \quad \; \: \, \, 
  (x\in \mathbb{R}). 
 \end{array}
 \right.\,\end{aligned}$$ Owing to Lady[ž]{}enskaja-Solonnikov-Ural’ceva [@lad-sol-ura] (see also [@lions]), the regularized problem (3.9) has a unique classical global solution in time $\phi_{\epsilon}=\phi_{\epsilon}(t,x)$ on $[\, 0,T\, ]\times \mathbb{R} $ for any $T>0$, that is, $$\phi_{\epsilon},\, \partial_x \phi_{\epsilon},\, 
\partial_x^2 \phi_{\epsilon},\, \partial_t \phi_{\epsilon}
\in C^{\infty }\bigl( \, [\, 0,\infty )\times \mathbb{R} \, \bigr)$$ because the equation in (3.9) is uniformly parabolic with variable coefficients. Further, the maximum principles (see [@ilin-kalashnikov-oleinik], [@pro-wei]) for (3.10) allows us to get the uniform boundedness to $\phi_{\epsilon}(t,x)$ as follows.

[**Lemma 3.1**]{} (uniform boundedness)[**.**]{}

Since $\phi^{\epsilon} _{0} \in H^{\infty}$, by using the above uniform boundedness and the standard arguments in the Sobolev space on the uniformly parabolic equations, we can see the classical $C^{\infty}$-solution $\phi_{\epsilon}$ also satisfies $$\phi_{\epsilon} 
\in C^{\infty }\bigl( \, [\, 0,\infty )\: ; \: H^{\infty} \bigr). $$ Then, for any fixed $\epsilon \in (0,1)$ and $T>0$, we can obtain the following a priori estimates which are depend upon $\epsilon$ and $T$ to the problem (3.9) as follows.

[**Lemma 3.2**]{} (a priori estimates I)[**.**]{}

[**Lemma 3.3**]{} (a priori estimates II)[**.**]{}

The proofs of Lemma 3.2 and Lemma 3.3 are given in the last part in this section. We can 