# Heat Equation PDE

Let $u(x,t)$ satisfy $u_{t}=u_{xx}$ for $x \in (0,1)$ and $t>0$, the boundary conditions $u(0,t)=u(1,t)=0$ for $t\geq 0$, and the initial condition $u(x,0)=f(x)$ for $x\in [0,1]$ with $f$ being a continuously differentiable function. Prove that
$\int_{0}^{1} |u(x,t)|^2 x\leq \int_{0}^{1} |f(x)|^2 x$, for any $t\geq 0$
Hint: use $2uu_{t}=(u^2)_t$ and $2uu_{xx}=(uu_{x})_{x}-2(u_x)^2$
Consider the function $g(t) = \displaystyle\int_0^1{\left(u(x,t)\right)^2dx}$ and show that it is not increasing! -differentiate.