Consider the advection-reaction equation in One-Dimension
$\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = u(1-u) + f(x,t); x\in\mathbb{R}, t>0$
with initial condition $u(x,0) = g(x)$, and $f, g$ are given nonnegative functions.
I'm searching for papers to give me some clue or proposition to ensure me just the existence of a solution for this class of hyperbolic problem, but I got nothing, how can I know if there is in fact a solution given an initial condition or not? Or someone could provide me with an article I haven't gotten it yet.