I am trying to solve the following problem from the 2021 Analysis of PDE Tripos exam:
Find all the characteristic curves to the equation $$ u_{t}-(x^2-1)u_{xx}=0. $$ When $x \neq 1,-1$, the definition given in lecture is that $\Sigma=\{\phi=0\}$ is the characteristic curve of the PDE if $$ (x^2-1)\phi_{x}^{2}=0. $$ we have that $\phi_{x}=0$. This $\phi(x,t)=f(t).$ Where $f: \mathbb{R} \to \mathbb{R}$ is some function.
When $x=1,-1$ the characteristic curve satisfies $$ \phi_{t}=0. $$ This $\phi(1,t)=g(x)$ and $\phi(-1,t)=h(x)$ where $g,h: \mathbb{R} \to \mathbb{R}$ are functions.
I'm not sure how to piece the 3 characteristic curves together. Any help would be greatly apprecaited. Thank you.