If you set up the one dimensional heat equation on an infinite line
$$\frac{\partial u}{\partial t} = \frac12 \frac{\partial^2 u}{\partial x^2}$$
knowing the initial conditions $u(x,0) = f(x)$ is enough to solve the equation. But if you have initial conditions along the time axis instead, $u(0,t)=g(t)$ this wouldn't be enough to solve the PDE. E.g. with $g(t)=0$, $u(x,t)= \alpha x$ is a solution for any $\alpha$. It seems like being given the first order space derivative, $u_x(0,t)=h(t)$ as well would be enough, because if we imagine $x$ is now the time variable and $t$ is the space variable, the PDE now depends on the second derivative of the time variable. Though I'm not sure if this is true.
But this is a problem more generally, and with some PDEs there isn't a clear time variable, so even this inuitive approach doesn't work in general for PDEs. So I would like to know whether, more generally, there is a way to tell what initial conditions are enough to specify a unique solution in linear PDEs.
This isn't a precisely stated problem, as I am not trying to solve a specific problem, so I don't know what form an answer to this question would take. But I would like to know any theorems or rules of thumb that can be used to tell when initial conditions are or are not sufficient for linear PDEs. In PDEs with a time dimension, this question usually translates to 'how much information about initial conditions do you need to predict what happens next', but I am also interested in PDEs without a time dimension.
