I have the following boundary value problem:
$$\nabla^2 T(x,y,z)=0$$
defined over $x\in[0,L], y\in[0,l], z\in[0,w]$ and subjected to
$$\frac{\partial T(0,y,z)}{\partial x}=\frac{\partial T(L,y,z)}{\partial x}=\frac{\partial T(x,0,z)}{\partial y}=\frac{\partial T(x,l,z)}{\partial y}=0$$
and
$$\frac{\partial T(x,y,0)}{\partial z}=A (T(x,y,0)-t_1)$$ $$\frac{\partial T(x,y,w)}{\partial z}=B (t_2-T(x,y,w))$$
$A,B$ are constants $>0$ and $t_1,t_2$ are functions of $T$.
Will the above system have a unique solution ?
Form of $t_1$ and $t_2$
$$t_1=e^{-b_c y/l}\left[t_{ci} + \frac{b_c}{l}\int_0^y e^{b_c s/l}T(x,s,0)ds\right]$$
$$t_2=e^{-b_h x/L}\left[t_{hi} + \frac{b_h}{L}\int_0^x e^{b_h s/L}T(x,s,w)ds\right]$$
$t_{hi}, t_{ci}, b_c, b_h$ are all constants greater than zero.