In studying PDE theory I started with the linear one-dimensional transport equation, which basically comes from the conservation law, $$ u_t + cu_x = 0,\,\,\,\,\,\,\,\,x\in \mathbb{R},\,t>0,$$ where $u=u(t,x)$ denote a density (i.e., mass per length unit). Keeping in mind the physical interpretation of the previous, if I add some (generic) initial condition $u(0, x) = g(x)$ I deal with an initial value problem $$\left\{\begin{matrix} u_t + cu_x = 0\,\,\,\,\,\,\,\,x\in \mathbb{R},\,t>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\ \\ u(0,x) = g(x)\,\,\,\,\,\,\,\,x\in \mathbb{R}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\\ \end{matrix}\right.$$ I know this is probably stupid to ask, but what is a rigorous definition of solution for the previous initial value problem? My doubts come from the fact that I don't know how to treat the line $t=0$ in terms of domain of definition for a solution.
Do I need to find $u\in C^1([0,+\infty)\times \mathbb{R})$ such that it satisfies both $(1)-(2)$, or just ask for a function $u\in C^1((0,+\infty)\times \mathbb{R})$ such that u satisfies $(1)$ and see $(2)$ as $$\lim_{t \to 0^{+}} u(t,x)=g(x)\,\,\,\,\,\,\,\,\, \forall x \in \mathbb{R}$$
I know that if $u=u(t,x)$ satisfies $(1)$ then, by method of characteristics, $u(t,x)=f(x-ct)$ on $ (0,+\infty)\times \mathbb{R}$ for an arbitrary $f:\mathbb{R}\longmapsto \mathbb{R}$. How do I have to use de IC $u(0,x)=g(x) \,\,\,\forall x \in \mathbb{R}$ to conclude that $u=g(x−ct)$ in the whole $[0,+\infty)\times \mathbb{R}$?
Thanks a lot in advance!!