Classify the following PDEs according to being elliptic, hyperbolic or parabolic. And for each one sketch the regions of ellipticity, parabolicity, and hyperbolicity
- Tricomi equation $$y\partial_{xx} u + \partial_{yy} u = 0,\qquad u = u(x, y);$$
- $$ x^2\partial_{xx} u + 2xy\partial_{xy} u + y^2\partial_{yy} u + \left(\partial_{x} u\right)^2 − e^u = 0$$
So I know that the first one is hyperbolic and the second is elliptic. but I don't quite know how to sketch them.
We use the notation $Au_{xx}+Bu_{xy}+Cu_{yy}$. We then have ellipticity when $\Delta := B^2-4AC<0$, parabolicity when $\Delta=0$, and hyperbolicity when $\Delta>0$.
For $y>0$, we have $\Delta = -4y<0$, so the equation is hyperbolic in this region. Similarly, we have a parabolic equation when $y=0$ and hyperbolic equation when $y<0$.
We have $\Delta = 4x^2y^2-4x^2y^2 = 0$ for all $x$ and $y$, so the equation is parabolic everywhere.