(i) Find $w:[0, \infty) \times \mathbb{R} \longrightarrow \mathbb{R}$ such that $w(t, \cdot)$ is a Schwartz function of $\xi$ for each $t$ and solves

$$\begin{gathered} w_{t}(t, \xi)+\left(1+\xi^{2}\right) w(t, \xi)=g(\xi), \\ w(0, \xi)=w_{0}(\xi), \end{gathered}$$

where $g$ and $w_{0}$ are given Schwartz functions and $w_{t}$ denotes $\partial_{t} w$. If $\mathcal{F}$ represents the Fourier transform operator in the $\xi$ variables only and $\mathcal{F}^{-1}$ represents its inverse, show that the solution $w$ satisfies

$$\partial_{t}\left(\mathcal{F}^{-1}\right) w(t, x)=\mathcal{F}^{-1}\left(\partial_{t} w\right)(t, x)$$

and calculate $\lim _{t \rightarrow \infty} w(t, \cdot)$ in Schwartz space.

(ii) Using the results of Part (i), or otherwise, show that there exists a solution of the initial value problem

$$\begin{gathered} u_{t}(t, x)-u_{x x}(t, x)+u(t, x)=f(x) \\ u(0, x)=u_{0} \end{gathered}$$

with $f$ and $u_{0}$ given Schwartz functions, such that

$$\|u(t, \cdot)-\phi\|_{L^{\infty}(\mathbb{R})} \longrightarrow 0$$

as $t \rightarrow \infty$ in Schwartz space, where $\phi$ is the solution of

$$-\phi^{\prime \prime}+\phi=f .$$