
Heat Equation PDE
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 $\displaystyle 2uu_{xx}=(uu_{x})_{x}2(u_x)^2$

Consider the function $\displaystyle g(t) = \displaystyle\int_0^1{\left(u(x,t)\right)^2dx}$ and show that it is not increasing! differentiate.