This is in part related to this question, but this is more general.
The Kirchhoff integral theorem says that given a solution $u = u(x, y, z, t)$ to the Helmholtz equation $\nabla^{2} u + k^{2} u = 0$, if we know the values of $u$ on a closed 2D boundary, we can use them to compute the value of $u$ at any point inside the surface using Kirchhoff's integral formula.
A similar property holds for the wave equation $\partial^{2}_{t} u = c^{2} \nabla^{2} u$.
Interestingly, not all PDEs seem to have such a property. Unlike the wave equation, the heat equation has the property that changing the solution in one local region can influence the solution in an arbitrarily far away region instantaneously. This means you can't compute $u(x, y, z, t)$ at a point purely by using values on any single 2D surface. You have to know $u$ on all points of $\mathbb{R}^{3}$ at time $t$ to find out what happens at time $t+\epsilon$, which defeats the purpose.
Question. What I'm wondering is, are there any other PDEs (time dependent or independent) such that you can compute the solution at a specific point simply by knowing the values of the solution on a 2D surface (at earlier times)?
Is there a way to characterize the class of PDEs that have this property? Are there references that discuss this?