We have a 1-d heat equation $\partial_t u = \partial_{xx} u$ with initial data $u(x,0) = f(x)$, where $f(x)$ is a $2\pi$ periodic and satisfies $\int_0^{2\pi}f(x)dx = 0$. The goal is to show $ \int_0^{2\pi}|u(x,t)|^2dx \leq c_1e^{-c_2t}\int_0^{2\pi}|f(x)|^2dx$.
I am given a hint to expand $u(x,t)$ as Fourier Series. However, I do not know how to deal with the integrand on the LHS which becomes a square of sum.
Any helps with this would be appreciated!
P.S. it is a problem set question. Yet I am taking this class with some prerequisites missing. I feel like that I am missing something here.
Hint: Consider the Fourier transformation $\hat u(.,t):\xi \mapsto \sum_{n \in \mathbb{Z}}c_n(t)\exp(in\xi/T)$.
Note that $\int_{0}^{2 \pi} |u(x,t)|^2dx = \text{constant} \times \int_{0}^{2 \pi} |\hat u(\xi,t)|^2d\xi = \text{constant} \times \left( \sum_{n \in \mathbb{Z}} |c_n(t)|^2\right)$ by the Parseval identity. Can you carry on from here knowing the functions $c_n(.)$?
P.S. A question where a similar approach can work is the following inequality: https://en.wikipedia.org/wiki/Wirtinger%27s_inequality_for_functions as well as the second answer here: https://math.stackexchange.com/a/619997/45470