I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset): \begin{equation*} u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^i(x,t)u_{x_i} + c(x,t)u = f, \quad \text{in $U\times (0,T)$} \end{equation*} and $u = 0$ on $\partial U \times [0,T]$ and $u=g$ on $U\times \{t=0\}$, admits a smooth (infinitely differentiable) classical solution.
According to L. C. Evans, Partial Differential Equations (1993), section 7.1.3, if $g, f$ are $C^\infty$, the coefficients $a^{ij}, b^i, c$ are $C^\infty$ and do not depend on $t$, then we have a smooth solution $u \in C^\infty(\bar{U}\times[0,T])$.
My question is: how important is the assumption that $a^{ij}, b^i, c$ are $t$ independent?
The case I'm particularly interested in is when $n=1$ with $U = (0,1)$, and $a^{ij}(x,t), b^i(x,t), c(x,t)$ are smooth in both $x, t$, but not constant in $t$ (i.e. $C^\infty([0,1]\times [0,T])$). What additional assumptions would be needed to guarantee the existence of a smooth classical solution $u$ to the above PDE?
One place I saw the independence on $t$ independenthypothesis being used was in the proof of Theorem 5: $\frac{d}{dt}\left[\frac{1}{2}\int_U\sum_{i,j}a^{ij}u_{m,x_i}u_{m,x_j}dx\right] = \int_U\sum_{i,j}a^{ij}u_{m,x_i}u_{m,x_j}'dx$. But if $a^{ij}$ $$ \frac{d}{dt}\left[\frac{1}{2}\int_U\sum_{i,j}a^{ij}u_{m,x_i}u_{m,x_j}dx\right] = \int_U\sum_{i,j}a^{ij}u_{m,x_i}u_{m,x_j}'dx$. But if $a^{ij} $$ is not independent of $t$, don't we just have an extra bounded term $\frac{1}{2}\int_U(a^{ij})'u_{m,x_i}u_{m,x_j}dx\leq \frac{1}{2}\max_{\bar{U}}|(a^{ij})'|\|u_m\|^2_{H^1_0(U)}$? $$ \frac{1}{2}\int_U(a^{ij})'u_{m,x_i}u_{m,x_j}dx\leq \frac{1}{2}\max_{\bar{U}}|(a^{ij})'|\|u_m\|^2_{H^1_0(U)}\;? $$ I don't fully understand every parts of Evans' proof, so I also hope any answer to this post would help guiding me as well.
