The solution to 1-D heat equation can be expressed via 1-D convolution. More formally, let $\frac{\partial u}{\partial t} = -k \frac{\partial^{2}}{\partial x^{2}}$, then we can find a solution $u = \int f(s)h_t(x-s)ds$.
However, when we consider the cartesian 3D coordinates ($\frac{\partial u}{\partial t} = -k \Delta u$), why cannot the solution be expressed as a convolution in 3 dimensions?
Thank you.