Heat equation, initial-boundary value problem

Question

Let $u (x, t)$ be a solution of the initial -boundary value problem $$\left\{\begin{array}{ll} U_t - U_xx = 0 & 0 < x < L, t > 0 \\ U (0, t) = U (L, t) = 0 & t > 0 \\ U (x, 0) = f (x) & 0 < x < L \end{array}\right.$$ Show that if there is a constant $C$ such that for $t > 0, |U (x, t)| <= Ce^ {-t^2}$, then $f (x) = 0$ for $0 < x < L$.

Answer 1

We may express $f$ as $$ f(x)=\sum_{k=0}^\infty a_k\sin \left(\frac{k\pi x}{L}\right). $$ Then $$ u(x,t)=\sum_{k=0}^\infty a_k\mathrm{e}^{-\frac{k^2\pi^2 t}{L^2}}\sin \left(\frac{k\pi x}{L}\right), $$ and hence $$ \lvert u(x,t)\rvert \le c\,\mathrm{e}^{-\frac{\pi^2 t}{L^2}}. $$ If $L=\pi$, then $$ \lvert u(x,t)\rvert \le c\,\mathrm{e}^{-t}. $$