I have a doubt about hyperbolic PDE's. The classical LeVeque's book, Numerical Methods for Conservation Laws, give us:
These notes concern the solution of hyperbolic systems of conservation laws. These are time-dependent systems of partial differential equations (usually nonlinear) with a particularly simple structure. In one space dimension the equations take the form $$\partial_t u+\partial_x f(u)=0$$ with $f'(u)$ diagonalizable, with real eingenvalues. (c.f. pages 1-2)
Well, many other locals has the following definition. Suppose the PDE
$$a(x_{1},x_{2}){\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+b(x_{1},x_{2}){\frac {\partial ^{2}u}{\partial x_{1}\partial x_{2}}}+c(x_{1},x_{2}){\frac {\partial ^{2}u}{\partial x_{2}^{2}}}+d(x_{1},x_{2}){\frac {\partial u}{\partial x_{1}}}+e(x_{1},x_{2}){\frac {\partial u}{\partial x_{2}}}+f(x_{1},x_{2})u=g(x_{1},x_{2})$$
If $b^2-4ac<0$, the PDE is elliptic.
If $b^2-4ac=0$, the PDE is parabolic.
If $b^2-4ac>0$, the PDE is hyperbolic.
However, the equation from LeVeque has $b=a=c=0$, so it would be parabolical...
Are there two kinds of hyperbolicity?
Many thanks in advance!