PDE Mathematical Behavior (1 Viewer)

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I am looking for an elegant way of demonstrating the parabolical behavior of the system:

[tex] \frac{\partial u}{\partial
x}+\frac{1}{r}\frac{\partial}{\partial r}(vr)=0

[tex] u\frac{\partial u}{\partial x}+v \frac{\partial
u}{\partial r}=\frac{1}{r}\frac{\partial}{\partial r}\Big(r
\frac{\partial u}{\partial r}\Big)

Any idea?. I have read some ways of doing so by establishing a complicated matrix of coefficients, but it is only valid for linear equations.

Thanks in Advance!

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