Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 29C / Solution

For positive time, convolution with the heat kernel givesDifferentiation under the integral verifies the heat equation, and the approximate identity property gives the initial value. Taking absolute values under the integral and using $e^{-|x-y{|}^2/(4t)}\le 1$ givesThe printed inequality omits the modulus even though $g$ is complex-valued. A complex number has no compatible order, so the displayed absolute-value estimate is the meaningful correction.

Substitution of $u=e^{it}v$ reduces the forced partial differential equation to $(i-\Delta )v=f$. With the Fourier transform convention for which $\widehat{\Delta v}=-|\xi {|}^2\hat{v}$, setThe denominator never vanishes for real $\xi$. Every derivative of its reciprocal has at most polynomial growth, so multiplying a Schwartz function by it gives another Schwartz function. The inverse Fourier transform therefore supplies the required $v$.

For the prescribed initial value the full solution is $u(t,x)=e^{it}v(x)+K_t\ast (g-v)(x)$. Applying the heat kernel estimate to the Schwartz function $g-v$ proves the stronger uniform conclusion