Let $d_1(t)$ and $d_2(t)$ be smooth functions from $[0,T]$ to $\mathbb{R}$. Suppose $L$ is a uniform elliptic operator and $u :\mathbb{R} \to \mathbb{R}$, we consider the problem \begin{equation} \left\{\hspace{5pt}\begin{aligned} &\dfrac{\partial u}{\partial t} - Lu = f(x,t)& \hspace{10pt} &\text{for $(x,t) \in \big(d_1(t),d_2(t)\big) \times (0,T]$} ;\\ &u(x,0) = g(x) & \hspace{10pt} &\text{for $\lambda \in \big(\lambda_1(T),\lambda_2(T)\big)$.}\\ &u(d_1(t),t) = p(t) & \hspace{10pt} &\text{for $t \in [0,T]$.}\\ &u(d_2(t),t) = q(t) & \hspace{10pt} &\text{for $t \in [0,T]$.} \end{aligned}\right. \end{equation} Here $f,g,p,q$ are some suitable functions.
Do we have some books or papers discussing some kinds of problems like this? I would like to have some existence and regularity results for this. I think it is a parabolic equation with domain depending on time. I have searched for "parabolic equation, time depending domain", "parabolic equation, moving boundary". But I do not have any result.