what is the Quasi Linear equation?

Question

Can anyone explain the Quasi Linear Equation

A first order PDE is called quasilinear if its coefficients depend on the variable u.

A example would help.

Answer 1

The definition you asked about works only in some specific cases, I think.

A quasi-linear equation, informally, is a PDE whose highest order terms appear only as individual terms, multiplied by lower order terms (I found this definition in this question). So the Burgers' equation is a quasi-linear first-order PDE because you can write it as $$(1, u) \cdot (u_t, u_x)=0 .$$

A more specific definition of quasi-linear PDE is the following. Let's first write a PDE in its general form: $$F(D^k(u), D^{k-1}(u), \ldots, Du, u, x) =0$$ where $x$ is in some open set $\mathcal{U} \subseteq \mathbb{R}^n$, $u: \mathcal{U} \to \mathbb{R}$ is the unknown function and $F: \mathbb{R}^{n^k} \times \mathbb{R}^{n^{k-1}} \times \ldots \times \mathbb{R}^{n} \times \mathbb{R} \times \mathcal{U} \to \mathbb{R}$ is given.

[$D^k(u)$ stands for the $k$-th multi-index derivative of $u$. That's a generalized way to write a derivative of a scalar multi-variable function.]

A PDE is quasi-linear if you can write it as: $$\sum_{|\alpha|=k} a_{\alpha} (D^{k-1} (u), \ldots, Du, u, x) D^{\alpha} (u) + a_0 (D^{k-1} (u), \ldots, Du, u, x) = 0.$$

Considering the Burgers' equation, we have $k=1$, $a_1 (Du, u, x) = (1, u)$ and $Du = (u_t, u_x)$. With a little bit of patience you can see that this extended definition actually works.

Source: L.C. Evans, Partial Differential Equations (2010), AMS