Partial Derivatives & the Gradient · Superchallenge problem

For constants $a$ and $b \ne 0$, the function $u(x,y,z,t) = t^{a}\exp\!\left(-\dfrac{x^2 + y^2 + z^2}{b\,t}\right)$ satisfies the heat equation $u_t = u_{xx} + u_{yy} + u_{zz}$ at every point with $t > 0$. Find $a + b$.