I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs:
Given a domain $[x_1, x_2] \times [t_0, t_1)$, find a function $u(t, x)$ which satisfies the following parabolic PDE with boundary conditions: $$ u_t+\mathcal{L}u = 0 \quad \text{ in } \quad (x_1, x_2) \times (t_0, t_1) $$ with the boundary condition $$ u(t, x) = f(t, x) \quad \text{ on } \quad {x_1} \times [t_0, t_1) \cup \{x_2\} \times [t_0, t_1) \cup [x_1, x_2] \times t_0. $$ Clearly, we assume that $\mathcal{L}$ is a parabolic operator and the function $f$ is nice enough.
I am interested in the existence/uniqueness result of the solution to such PDE. Almost all papers/authors refer to Chapter 3 of the book "PDEs of Parabolic Type" by A. Friedman. However, Friedman has a much more general setup, which I am not sure is needed in the above problem (due to a very simple domain). In addition, I find Friedman's book quite hard to read.
I would greatly appreciate it if anyone would recommend me a reference where the existence/uniqueness results for the above PDE are written down. If I understand correctly, another classical reference in PDE literature is Evan's book. However, I think Evans' book has only a theorem about the existence/uniqueness of the above PDE if the boundary condition is zero, meaning that $f(x_1, t) = f(x_2, t) = 0, \forall t \in (t_1, t_2)$ (this is 7.1.2 in Evans' book). The same simplified boundary condition can be found in many other books, but I cannot find even one more reference for the general case (except for Friedman's book).
Suppose you can find $F : [x_1, x_2] \times [t_0, t_1) \to \mathbb{R}$ such that $F|_{\partial([x_1, x_2]) \times [t_0, t_1)} = f$. Then we can write $u = F + v$, where $v$ should satisfy the PDE $$v_t - Lv = -F_t - LF$$ $$v|_{\partial([x_1, x_2]) \times [t_0, t_1)} = 0.$$ $$v(t_0,x) = f(t_0, x) - F(t_0, x).$$ This equation has an explicit formula for the solution $v(t)$ in terms of eigenfunctions of $L$ that vanish on $\partial[x_1, x_2]$. See excercises 4-10 on page 489 of "Partial Differential Equations I Basic Theory" by Taylor for estimates on the solution.
Then it remains to find an $F$ and see how smooth $F$ is in terms of $f$. That should be simple since we are in 1D. Compare proposition 1.7 on page 320, which shows how you can get $F(t)$ from $f(t)$. If $E$ denotes an extension map from $\{x_1, x_2\}$ to $[x_1, x_2]$, which could be obtained from proposition 1.7 by a partition of unity argument, we can use $F(t) = Ef(t)$.