Take $$ s\frac{\partial^2}{\partial s^2}\phi_s(x)=-x\frac{\partial}{\partial x}\phi_s(x) \tag{1}$$
which is well posed forward in time under the right conditions, $\forall s >0$ and ill posed backward in time where in finite time we get a singularity in the form of some distribution function that can be found by taking a limit on a certain solution of $(1)$. I will deal strictly with one particular solution to the equation (which makes things well posed).
The Riemannian foliation with standard Euclidean metric will be employed as a solution to $(1)$.
$$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{s}{\log x}}:s \in (0,\infty) \bigg\rbrace$$
where $x\in(0,1)$.
Take the set of nested pairings, the parings will be denoted $\S$, of solutions $M_{s,1/s}=\mathrm{Nest}\lbrace\phi_s~ \S ~\phi_{1/s}\rbrace$. Note that $s=1$ recovers the identity leaf $\phi_1$. Define an equivalence relation s.t. the boundary leaf $\phi_s$ is glued to $\phi_{1/s}$ for each $s$. We get shells of topological manifolds (sphere-like) and say we embed the singular points above $x=1$ and $x=0$ at $q=(1,0,1/2)$ and $p=(0,1,1/2)$.
Let $M$ be pre-solutions i.e. abstract topological non rigid surfaces, set up as pre-sections which we'll want to deform into rigid analytic objects s.t. the fibers on the pre-solutions project directly onto the real analytic leaves of the foliation $\phi_s$.
So $(1)$ comes into play where rotating a fiber cyclically around $M$ corresponds to deforming a planar solution $\phi_{1/s}(x)$ forward in time until it reaches $\phi_s(x)$ then deforming it backward it time until it reaches $\phi_{1/s}(x)$. This oscillatory solution lifts onto the pre-solution $M$.
With the gluing of solutions are there resources/references I can read that show how to obtain a diffusion PDE on the pre-solutions whose fibers project onto the planar solutions to $(1)$? I think the PDE on the pre-solutions will be very similar but have an essential rotational aspect.
Related question I found after writing this question: Gluing together solutions to differential equations.
