Suppose $U=\Omega \times (0,\infty),$ $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $u \in C^{2,1}(\overline{U})$ satisfies $u_t=\Delta u - u^3$ in $U$ with $u(x,t)=0$ on $\partial \Omega \times (0,\infty).$ Construct an energy functional and use it to show that $u(x,t) \rightarrow 0$ as $t \rightarrow \infty.$
I haven't learnt constructing such a functional. I'm studying for my preliminaries. Any help is much appreciated.
If $E[u](t)=\int_\Omega |u(x,t)|^2 dx$ then
$$\frac{dE}{dt}=2 \int_\Omega u(x,t) u_t(x,t) dx = 2 \int_\Omega u(x,t) (\Delta u(x,t) - u(x,t)^3) dx \\ = -2 \int_\Omega |\nabla u(x,t)|^2 dx - 2 \int_\Omega u(x,t)^4 dx \leq 0$$
with equality iff $u(\cdot,t)=0$ (strictly speaking if $u(x,t)=0$ for a.e. $x$, but you have enough regularity that this just means $u(\cdot,t)=0$).