consider the heat equation on a circle $\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$ with initial data $f(x)$ integrable on $[0,1]$.
Let $u(x,t)=\sum_{-\infty}^{\infty}a_ne^{-4n^2\pi^2t}e^{2\pi in x}$, where $a_n$ is the fourier coefficient of $f$. I know $u(x,t)$ solve the heat equation.
I am trying to show that $||u(x,t)-f||_2\to 0$ as $t\to 0$. I try to use the Parseval's identity since here we have the information of fourier coefficients and I got $||u(x,t)-f||_2=\sum_{-\infty}^{\infty}|a_n(e^{-4\pi ^2n^2t}-1)|^2$. I am trying to estimate $e^{-4\pi ^2n^2t}-1$ and I guess it is impossible to get $\sigma>0$ such that $|e^{-4\pi^2n^2 t}-1|<\epsilon$ for all $|t|<\sigma$ independent of $n$, if we can get such $\sigma$, then the problem will be settled. Any suggestions?
\begin{align} u(x,t)-u(x,0) &= \sum_{n=-\infty}^{\infty}a_n(e^{-4n^2\pi^2 t}-1)e^{2\pi inx} \\ \int_{0}^{1}|u(x,t)-u(x,0)|^2dx &= \sum_{n=-\infty}^{\infty}|a_n|^2|e^{-4n^2\pi^2 t}-1|^2 \end{align} The above tends to $0$ as $t\downarrow 0$ by the Lebesgue Dominated Convergence applied to the last sum on the right.