# Classification of PDEs with respect to nonlinearity

Firstly, I know this question was asked sometimes on Mathstack, but I don't sure if I understood the classification by the answers that I read here. I'm studying by myself PDEs and I'm trying understand how classifying PDEs with respect to them nonlinearity. According a lecture note that I found on the internet, PDEs are classified with respect to them nonlinearity as follows:

A partial differential equation is a mathematical equation involving partial derivatives, which has the form

$$(1) \ F \left( x_1, \cdots, x_n, u, \frac{\partial u}{\partial x_i}, \frac{\partial^2 u}{\partial x_i \partial x_j}, \cdots, \frac{\partial^m u}{\partial x_{i_1} \cdots \partial x_{i_m}} \right) = 0,$$

where $$x = (x_1, \cdots, x_n)$$ belongs to some domain $$\Omega \subset \mathbb{R}^n$$ and $$u: \Omega \longrightarrow \mathbb{R}$$ is a function with partial derivatives until order $$m$$. A PDE has order $$m$$ if the highest order of the partial derivative is $$m$$.

We say that the PDE $$(1)$$ is linear if $$F$$ is linear with respect to $$u$$ and all its partial derivatives, otherwise, the PDE is nonlinear and it is classified as follows:

1. Semilinear equations: $$F$$ is non linear with respect to $$u$$, but it is linear with respect to all its partial derivatives;

2. Quasilinear equations: $$F$$ is linear only with respect to the partial derivatives which order is the same of the PDE;

3. Fully nonlinear equations: $$F$$ is non-linear with respect to the partial derivatives which order is the same of the PDE.

I don't know how to interpret linearity in this definition, because, for example, consider the inviscid Burgers' equation:

$$F(u, u_t, u_x) := u_t + u u_x = 0$$

By that I understood of the definition, given $$u, v$$ two solutions for the PDE $$F$$, we see that

$$F(u + v, u_t + v_t, u_x + v_x) := (u + v)_t + (u + v) (u + v)_x = u_t + u u_x + v_t + v v_x + u v_x + v u_x = F(u, u_t, u_x) + F(v, v_t, v_x) + u v_x + v u_x$$

Thus,

$$F(u + v, u_t + v_t, u_x + v_x) \neq F(u, u_t, u_x) + F(v, v_t, v_x),$$

then the inviscid Burgers' equation can't be quasilinear, indeed, the PDE would be fully nonlinear, but this PDE is quasilinear according the author of the lecture notes. By an analogous computation, we have the same problem when we consider Korteweg-de-Vries (KdV) equation:

$$F(u) := u_t + u u_x + u_{xxx} = 0.$$

By the other hand, Avner Friedman defines quasi-linear and semi-linear equations of second order on page $$187$$ of his book about PDEs of parabolic type as follows:

$$(1.3) \ \sum_{i,j = 1}^n \Phi_{ij} (x,t,u, \nabla_x u) \frac{\partial^2 u}{\partial x_ \partial x_j} - \Phi_0 (x,t,u, \nabla_x u) \frac{\partial u}{\partial t} = f(x,t,u, \nabla_x u),$$

where $$\nabla_x u = \left( \frac{\partial u}{\partial x_1}, \cdots, \frac{\partial u}{\partial x_n} \right)$$ are called quasi-linear parabolic equations.

equations of the form $$(1.3)$$ for which $$\Phi_0 \equiv 1$$ and the $$\Phi_{ij}$$ are functions of $$(x,t)$$ only. These equations are called semilinear equations.

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva defined quasilinear equation of general form of the same way of Friedman defined.

I interpreted an equation here as quasilinear if the coefficient which multiply the partial derivative of the same order of the PDE as a function which depends on $$x, t$$, but depends on $$u$$ and the partial derivatives with the lower order as the order of the equation too!

According this interpretation, inviscid Burgers' equation would be quasilinear, which matches with the example of the author of the lecture notes, but the Korteweg-de-Vries (KdV) equation wouldn't be quasilinear since the coefficient which multiply $$u_{xxx}$$ (I will denote this coefficient by $$g$$) is such that $$g \equiv 1$$, then $$g$$ doesn't depend on $$u$$ and the partial derivatives with the lower order as the order of the equation, which doesn't match with the example of the author of the lecture notes.

$$\textbf{P.S.:}$$ while I was writting this post, I realized that the example with KdV equation doesn't contradicts the Friedman's definition since his definition is $$\textbf{only for PDEs of second order}$$ and KdV equation has order $$3$$, but I feel that I don't understand well the classification of PDEs according to nonlinearity, so I would like some explanation in order to clarify the definition for me.