I'm currently dealing with the heat equation, but am having some issues. In particular, the following:
Let $f(x,t)$ be a solution to the heat equation $\frac{\partial f}{\partial t} = k \frac{\partial^2 f}{\partial x^2}$, s.t. $k>0$, $f(x,0) = f_0(x)$ is continuous in $[0,1]$, and $f_0(0) = f_0(1)$. Given $\int_0^1f_0(x)dx = 0$, find and prove $\lim_{t\to\infty}e^{kt}f(x,t),$ and describe the convergence (pointwise/$L^2$/uniform).
I would greatly appreciate some help!