Suppose $u$ is a solution of the heat equation with the property that $|\int\limits_{-\infty}^{\infty}u(x,0)dx| < \infty$, and
$u_{x}(x,t) \rightarrow 0$ as $x \rightarrow \pm \infty$. Then integrating the PDE, we find
$$\frac{d}{dt}\int\limits_{-\infty}^{\infty}u(x,t)dx = 0 \quad (1)$$
so that thet total heat energy is conserved:
$$\int\limits_{-\infty}^{\infty}u(x,t)dx = \text{ constant.} \quad (2)$$
Could someone explain how to get from the initial conditions to equation $(1)$?
I was thinking to use Leibniz integral rule, but got stuck at the following stage:
$$\frac{d}{dt}\int\limits_{-\infty}^{\infty}u(x,t)dx = \int\limits_{-\infty}^{\infty}u_t(x,t)dx \quad + \quad 0$$
as I see no mention on how the above should behave at the boundaries.
The notes can be found online at IITD notes on The Heat Equation.
Best Answer
$\int_{-\infty}^\infty u_t(x,t) dx = \int_{-\infty}^\infty u_{xx}(x,t) dx = u_x(\infty,t)-u_x(-\infty,t)$. Now you use your decay condition. (The more interesting question is to prove that decay condition a priori.)