# Solving time dependent boundary conditions heat PDE

I am attempting to solve the heat equation $$\frac{\partial T}{\partial t}=\nabla^2T$$, where $$T=T(x,y,z,t)$$, subject to the following boundary conditions:

$$\frac{\partial T}{\partial x}|_{x=10}=\frac{\partial T}{\partial x}|_{x=-10}=0$$

$$\frac{\partial T}{\partial y}|_{y=10}=0$$

$$T(0,0,z_{crit}(t),t)=f(t)$$ where $$z_{crit}(t)=t$$ and $$f(t)=e^{-t}$$

$$\frac{\partial T}{\partial y}|_{y=0}=\frac{\partial T}{\partial z}|_{z=0}=0$$

But my attempted code has not worked properly and I encounter frequently many different errors.

Is there any way to fix this issue?

f[t_] := Exp[-t]
eqns = {
D[c[x, y, z, t], t] == Laplacian[c[x, y, z, t], {x, y, z}],
Derivative[1, 0, 0, 0][c][10, y, z, t] == 0,
Derivative[1, 0, 0, 0][c][-10, y, z, t] == 0,
Derivative[0, 1, 0, 0][c][x, 10, z, t] == 0,
c[0, 0, t, t] == f[t]
}