Let \begin{equation} a(t,x,u) \frac{\partial u}{ \partial t} + b(t,x,u) \frac{\partial u}{ \partial x} - c(t,x,u) = 0, \\ u(0,x) = \varphi(x), \end{equation} be a quasilinear, $2$-dimensional first oder PDE with $a,b,c \in \mathcal{C}^{1}(\mathbb{R}^3, \mathbb{R})$ and $\varphi \in \mathcal{C}^{1}(\mathbb{R}, \mathbb{R})$.
Under which conditions does this PDE have a unique classical global solution $u \in \mathcal{C}^{1}(\mathbb{R}^2, \mathbb{R})$ ? And when does it have a unique local solution? How is the situation in higher dimensions?
We have a theorem due to Cauchy-Kovalevsky, which gives existence and uniqueness for a particular class of first order PDE which is stated as follows: Let $u_{t}=F(t,x,u,u_{x})$ and $u(0,x)=\phi(x)$. Suppose that $\phi$ is analytic in the neighborhood of origin and $F$ is analytic in the neighborhood of $(0,0,\phi(0),\phi'(0))$, then the PDE with initial data has solution which is analytic around origin of $\mathbb{R}^{2}$ and unique in class of analytic functions. Also by using method of characteristic or Lagrange method you can convert your PDE into system of ordinary differential equations and can apply existence and uniqueness theorems from ordinary differential equations.