 Solution

For fixed frequency $\omega$, the Fourier transform of a derivative gives $\widehat{u_{xx}}=-{\omega }^2\hat{u}$. The transformed initial value problem is thereforeSolving this scalar ordinary differential equation yieldswhere the value at $\omega =0$ is interpreted by continuity and equals two. Thus diffusion suppresses high frequencies by the Gaussian multiplier while preserving the zero-frequency mass.

The differentiation and transform steps are justified for the heat kernel solution $u=q_t\ast T$: for positive time its spatial derivatives are integrable, since Gaussian derivatives are integrable and $T$ is bounded with compact support. Alternatively, the transformed calculation first suggests the formula, and the next part verifies the resulting solution directly.