When solving the heat equation on say $\mathbb{R}$ (or $[0,2\pi]$, whichever is easier to talk about) we are posing Cauchy data on the surface $t=0$. My understanding is that $t=$constant are precisely the characteristic surfaces of the heat equation.

I realize this question may seem elementary to those who know the answer, but I'm confused as to how this makes sense. I understand how to solve the heat equation using Fourier series (on $[0,2\pi]$) or the fundamental solution on $\mathbb{R}$ but I thought we were not able to pose Cauchy data on characteristic surfaces? Is the point here that we are not solving the equation using the characteristics? Shouldn't $u_t$ be automatically specified in terms of what $u_{xx}(t=0,x)$ is?