Consider the semilinear PDE $$(2+y^2)u_{xx}+y^2u_{yy}+2z^2u_{zz}=0$$ and we are asked to classify it. The eigenvalues of the corresponding matrix $$\left( \begin{array}{ccc} 2+y^2 & 0 & 0\\ 0 & y^2 & 0\\ 0 & 0 & 2z^2 \end{array} \right)$$ are $2+y^2,~y^2,~2z^2$, so if $yz\neq 0$ then all are positive and it is elliptic. If $y=0,~z\neq 0$ or $z=0,~y\neq 0$ then it is parabolic.
In the case $y=z=0$, what is the name of the type of the equation?
Thank you in advance.
I am not a mathematician, but the equation can be put into the form: $$-\mathrm{div}[\vec{F}(\vec{u})]+\vec{v}(\vec{x})\cdot\vec{\mathrm{grad}}u=0$$ Which has a physical meaning.
Where: $$\vec{F}=\mu\, \vec{\mathrm{grad}}u=\left[\begin{array}{cc} (2+y^2) & 0 & 0\\0&\frac{y^3}{3}&0\\0&0& \frac{2z^3}{3} \end{array}\right]\left[\begin{array}{c} \partial_xu\\ \partial_yu\\\partial_zu \end{array}\right]\qquad \vec{v}(\vec{x})=\left[\begin{array}{c}0\\y^2\\2z\end{array}\right]$$ If the entries of $\mu$ are strictly positive or $0$ and greater than terms grouped in $S$ the equation considered has a dominant elliptic behaviour. If the same occurs but $S$ is dominant it has a hyperbolic dominant behaviour.
The case in which some of the entries of $\mu$ are negative is unphysical and I don't know how must they be treated mathematically.
Hope this helps
Unles, I'm not interpreting something correctly, for $y=z=0$, your equation degenerates to
$$ u_{xx}=0 $$ which is the basic one-dimensional diffusion equation which is notionally elliptic, although in one dimension, it would really be a two-point boundary value problem.
