Let $W(t,x,y)$ be a function that satisfies the partial differential equation
$$\frac{\partial W}{\partial t}+ a\frac{\partial^2 W}{\partial x^2} + b \frac{\partial^2W}{\partial y^2} + c\frac{\partial W}{\partial x} + d\frac{\partial W}{\partial y}+ eW = 0$$
If we are given that the "final condition" $W(T, x, y) = f(x,T)$ does not depend on $y$, can I deduce (or under which conditions) that $$\frac{\partial W}{\partial y} = 0$$?