Prove that if $f$ is a solution of the heat equation.

Question

Let $M^{k}\subset\mathbb{R}^{n}$ be a compact, oriented manifold, and assume that $f:M^{k}\times[0,\infty)\to\mathbb{R}$ is smooth. The heat equation is $$\triangle_{x}f(x,t)=\dfrac{\partial f(x,t)}{\partial t}$$

Prove that if $f$ is a solution of the heat equation satisfying $f(x,0)=0$ for all $x\in M^{k}$ and $f(y,t)=0$ for all boundary points $y\in\partial M^{k}$ and all times $t\in [0, t_{0}]$, then $f$ vanishes identically on the set $M^{k}\times [0,t_{0}]$.

Any idea or approach of the problem. Thanks!

Answer 1

Consider

$$E(t) = \frac 12 \int_M f (x, t)^2 d\mu(x).$$

Then

$$\begin{split} \frac{d}{dt} E(t) &= \int_M ff_t d\mu \\ &= \int_M f \Delta f d\mu \\ &= -\int _M \| \nabla f\|^2 d\mu\le 0. \end{split}$$

(Note that we used integration by part, the boundary term vanish as $f|_{\partial M} = 0$). Thus $E(t)$ is nonincreasing. Since $E(0) = 0$, then $E(t) = 0$ and so $f$ is identically zero.