Problem. Consider the PDE $$ u_t + u_x = u_{xx}, \qquad (t,x) \in (0,+\infty) \times (0,1). $$
(i) Write the unique solution $\overline{u}=\overline{u}(x)$ which does not depend on time and is of class $C^2([0,1])$ satisfying $u(0)=1$, $u(1)=0$.
(ii) Show that all other solutions $u=u(t,x) \in C^2([0,+\infty) \times [0,1])$ with $u(t,0)=1$ and $u(t,1)=0$ converge to $\overline{u}$ in the $C^0$ norm as $t \to +\infty$.
Part (i) is indeed very easy: since $\overline{u}$ is independent on time, we have to solve the boundary value problem $$ \begin{cases} u^{\prime\prime} - u^\prime = 0\\ u(0)=1 \\ u(1)=0 \end{cases} $$ whose solution is $$ \overline{u}(x) = \frac{e}{e-1} - \frac{1}{e-1}e^x. $$
Now I do not know how to solve point (ii). I have tried a change of variables $(t,x) \mapsto (t^\prime, y)$: in particular, setting $y=x-t$ and $t^\prime=t$ I have obtained a heat equation $u_t - u_{yy}=0$ but I do not know how to handle the boundary conditions. I think there is something hidden in the text that I cannot see... Could you help me, please?
Thanks.