Consider the following general form of a quasilinear parabolic PDE $$ u_t = a(x,u,u_x)u_{xx} + b(x,u,u_x) \ \ \textrm{ for }-1<x<1, \tag{1}$$ with inhomogeneous boundary condition $u_x(x,t) = g(u(x,t))$ for $x=\pm 1$.

I am looking for references that shows the following: for any initial data $u_0$ living in some suitable Holder space, there exists a unique local-in-time classical solution $u$ in some time interval $0\le t<T$, in the sense that $u,u_x$ are continuous in $[-1,1]\times [0,T)$, $u_{xx}, u_t$ are continuous in $(-1,1)\times (0,T)$ and $u$ satisfies equation (1) in $(-1,1)\times (0,T)$.

I knew of a paper by Lunardi: "Abstract Quasilinear Parabolic Equations" (See here https://doi.org/10.1007/BF01456097) that proves the claim by establishing some a-priori gradient estimate. Correct me if I am wrong, but it seems to me that it only works for zero Neumann condition. If it helps, the equation I had in mind actually stems from curvature driven flow in a two-dimensional domain.