Pierre-Gilles Lemarie-Rieusset, *The Navier-Stokes Problem in the 21st Century* treats the heat equation on $\mathbb{R}^3$ for time $t\geq 0$, and proves uniqueness of suitably smooth solutions by a kind of energy argument. For a solution $u(t, x)$ with $u(0,x)=0$ he looks at the integral over all $x\in \mathbb{R}^3$  

\begin{equation}\int |u(t,x)|^2 e^{-|x|}\,dx.\end{equation}
Using the equation $u\_t=\Delta u$, calculations very like other energy proofs show
\begin{equation}
 \frac{d}{dt}\int |u(t,x)|^2 e^{-|x|}\,dx = -2\int |\vec{\nabla} u|^2 e^{-|x|}\,dx\ + 2\int u e^{-|x|} \sum\_{j=1}^{3}\frac{x\_j}{|x|}\partial\_j u\,dx. 
\end{equation}
Those calculations begin with integration by parts and proceed by pretty straightforward calculus. Lemarie-Rieusset immediately concludes
\begin{equation}
 \frac{d}{dt}\int |u(t,x)|^2 e^{-|x|}\,dx \leq \frac{1}{2}\int |u(t,x)|^2 e^{-|x|}\,dx.
\end{equation}
But I do not see how.



Unique solutions to the heat equation on $\mathbb{R}^3$
Sorry, maybe my previous comment was not clear enough. You have $$\frac{d}{dt} \int |u|^2 e^{-|x|}\, dx=-2\int |\nabla u|^2 e^{-|x|}\, dx +2 \int u \nabla u \cdot \frac{x}{|x|}e^{-|x|}\, dx.$$
Now use $2|u \nabla u| \leq \frac 12 |u|^2+2|\nabla u|^2$
to estimate the last integral.


