I am researching a system of pdes and it leads me to study the classical solution of one dimensional linear parabolic equation: $u_t+Lu=f,\, t\in[0,T],x\in \Omega=[0,1]$, where $L$ is non-divergence elliptic operator $L=-a(x,t)u_{xx}+b(x,t)u_x+c(x,t)u$. The initial condition is $u(x,0)=0$, the mixed boundary condition is $u(0,t)=0, \, u_x(1,t)=0$. Suppose that $a(x,t)>0$ on $[0,T]\times \Omega$ and $a(x,t), b(x,t), c(x,t), f(x,t)$ are smooth enough.
My question is that, is there any work claiming the well-posedness in classical sense (that is $u\in C^{1,2}([0,T]\times \Omega)$) of this problem and giving a priori estimate for u in (probably) some Holder space frame work? If there is no such research on that kind of mixed boundary condition, then the results on Neumann or Dirichlet could also be helpful for the first step of reading.
I have found some works from DuChateau and Muzylev but they work only for specific $L$.
I am a quite new learner at the solution in classical sense, can you please help me, from where should I read about it? Thank you!
