Unique solutions to the heat equation on $\mathbb{R}^3$

Question

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.

Answer 1

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.