In our previous discussion on strictly linear first-order partial differential equations, we observed that the unknown function $u$ and its partial derivatives $u_x$ and $u_y$ appeared only to the first power with coefficients depending solely on the independent variables $x$ and $y$. However, many physical phenomena, such as gas dynamics and traffic flow, are governed by equations where the rate of change depends on the state of the system itself. This leads us to the study of Quasi-Linear and Semilinear PDEs, which bridge the gap between simple linear models and fully nonlinear systems.

A first-order PDE is called quasi-linear if it is linear in the first-order partial derivatives $u_x$ and $u_y$, but the coefficients of these derivatives may depend on the independent variables $x, y$ and the unknown function $u$. The standard form is given by the equation $$\begin{equation} a(x, y, u)u_x + b(x, y, u)u_y = c(x, y, u) \cdots\text{(2.1)}\nonumber\end{equation}$$ where $a, b,$ and $c$ are continuous functions of their arguments.