In my PDE class, my instructor wrote the following notes:
Consider equations $u_t - \Delta u = 0$ in $\Omega$, where $\Omega \subset \mathbb{R}^n$ is bounded. Suppose boundary conditions $u = u_0(x)$ at $t=0$, and $u=0$ at $\partial \Omega$.
Multiplying the equation by $u$ and integrating by parts gives
$$\frac12 \frac{d}{dt} \int_{\Omega} u^2 + \int_{\Omega}|\nabla u|^2 = 0$$
(with no boundary terms, using the bc). This already shows
$$\frac{d}{dt} \int_{\Omega}u^2 \leq 0 \tag{1}$$
which gives uniqueness, since the problem is linear (so the difference of two solutions has initial data 0).
My Questions:
- I'm not seeing the integration by parts here, can someone talk me through it?
- How are we deducing (1)?
- How does (1) show uniqueness?
Thanks for your help, the details are eluding me here.