Let $\displaystyle u(x,t)$ satisfy $\displaystyle u_{t}=u_{xx}$ for $\displaystyle x \in (0,1)$ and $\displaystyle t>0$, the boundary conditions $\displaystyle u(0,t)=u(1,t)=0$ for $\displaystyle t\geq 0$, and the initial condition $\displaystyle u(x,0)=f(x)$ for $\displaystyle x\in [0,1]$ with $\displaystyle f$ being a continuously differentiable function. Prove that

$\displaystyle \int_{0}^{1} |u(x,t)|^2 x\leq \int_{0}^{1} |f(x)|^2 x$, for any $\displaystyle t\geq 0$

Hint: use $\displaystyle 2uu_{t}=(u^2)_t$ and $\displaystyle 2uu_{xx}=(uu_{x})_{x}-2(u_x)^2$