Solutions to the heat equation converge to the similarity solution

The heat equation $$ \begin{align} u_t &= u_{xx}, \quad x \in \mathbb{R}, \quad t > 0, \\ u(x, 0) &= u_0(x) \end{align} $$ admits a self-similar solution of the form $$ K(x, t) = \frac{1}{(4 \pi t)^{1/2}} \exp\left(\frac{-x^2}{4t}\right) $$ when the initial …