5. Suppose that $u$ is a solution of
\begin{cases}
u_t = x u_{xx} + u_x, & t \ge 0, 0 \le x \le 1, \\
u(x, 0) = \sin(\pi x), & 0 \le x \le 1, \\
u(0, t) = u(1, t) = 0, & t \ge 0.
\end{cases}
Use the energy method to prove that
$\int_0^1 u^2(x, t) dx \le \frac{1}{2}$ for any $t \ge 0.$
