the smoothing effect

Take a jagged, spiky temperature profile — even one with corners or jumps — and let the heat equation act on it for the briefest instant. The jaggedness is gone. The moment time moves forward, the profile becomes perfectly smooth, with derivatives of every order, no matter how ragged it started. This is the smoothing (or regularizing) effect of diffusion.

Concretely, the Cauchy solution u(x,t) = integral of G(x - y, t) f(y) dy is a convolution of the data f with the heat kernel G. The kernel G is a Gaussian, which is C-infinity (infinitely differentiable) and in fact real-analytic. Convolution inherits the smoothness of the smoother factor: even if f is merely bounded, or has jumps, the integral against an infinitely smooth Gaussian is infinitely smooth in x for every t > 0 — and analytic too. You can also see it through Fourier modes: rough data is rough because it contains large high-frequency components, and the heat equation kills the mode of frequency n at rate e^(-k n^2 t). For any t > 0 those high-frequency parts are exponentially crushed, leaving a smooth function. The smoothing is instantaneous: there is no waiting period, smoothness appears at every positive time.

This is one of the deepest distinctions among PDE types. Parabolic equations smooth; hyperbolic equations like the wave equation do not — there, singularities in the data propagate forever along characteristics and a kink stays a kink. The smoothing effect underlies the heat equation's use as a denoiser and image blur, its irreversibility (you have thrown away the fine detail, so you cannot reconstruct the rough past), and the regularity theory of parabolic PDE. Caveat: smoothing is a forward-in-time blessing and a backward-in-time curse — because information about fine structure is destroyed, the backward heat equation is ill-posed.