Quasilinear parabolic equations with nonlinear monotone boundary conditions

Of concern is the following quasilinear parabolic equation with a nonlinear monotone boundary condition: $$ \cases \displaystyle u_{t} (x, t) = \frac{\partial \alpha (x, u_{x})}{\partial x} + g(x, u), \quad (x, t) \in (0, 1) \times (0, \infty), \\ (\alpha (0, u_{x}(0, t)), - \alpha (1, u_{x}(1, t))) \in \beta (u(0, t), u(1, t)), \\ u(x, 0) = u_{0}(x). \endcases \tag $*$ $$ Here $ \beta $ is a maximal monotone graph in $ {\mathbb R} \times {\mathbb R}$, which contains the origin $(0, 0)$. It is showed that (*) has a unique strong solution $ u $, with the property that $$ \sup_{t \in [0, T]}\|u(x, t)\|_{C^{1+ \nu}[0, 1]} $$ is uniformly bounded for $ 0 0 $