Heat-kernel convergence for Gaussian-integrable initial data ($K_t\ast \phi \to \phi$)

Suppose $\phi$ is continuous at $x$ and, for some $T>0$,For $0<t<T$, its convolution with the heat kernel $K_t(s)=(4\pi t{)}^{-1/2}{e}^{-s^2/(4t)}$ is smooth and solves the heat equation. On every compact set in $\mathbb{R}\times (0,T)$, spatial or time derivatives of the heat kernel are bounded by the displayed integrable weight times a constant: their polynomial factors are absorbed by the strictly narrower Gaussian. This justifies differentiation under the integral sign.

Moreover, $\int K_t=1$, soInside $|\xi -x|<\delta$, continuity makes the difference uniformly small. Outside that interval, for $t\le T/4$, the inequality ${\xi }^2\le 2(\xi -x{)}^2+2x^2$ givesThe right side tends to zero, proving that the weighted tail contribution vanishes; the constant term $\phi (x)$ is controlled by the vanishing Gaussian tail mass. Thus $(K_t\ast \phi )(x)\to \phi (x)$ as $t\downarrow 0$. In particular, continuous data of polynomial growth satisfy this hypothesis, even when they are unbounded. This extends the Gaussian approximate identity argument beyond bounded data.