This is an extension (and more distilled version) of Extension of PDE's to critical strip, with new information. I am fairly sure that my constructions are an alternate description of the De Brujn constant method. Specifically I think $\Lambda=0$ corresponds to $\phi_0.$ This is what my question will be about at the end.
Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed.
$$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$
for which we have our pre-assigned unique solution:
$$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$
We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively.
We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay criterion, and we know its Mellin transform exists and has a closed Bessel form. This will be employed to extract a 1-parameter family of functional equations for the Riemann zeta function.
$$\Gamma(\phi,r,s)\cdot \zeta(s) = \Gamma(\hat{\phi},r,1-s)\cdot \zeta(1-s)$$
whereby moving $\zeta(s)$ to the RHS gives us a partial differential relation on the factor $\Gamma(\phi_s,r)$
$$r^2 \frac{\partial ^3}{\partial r^3}\Gamma(\phi_s,r) =s^2 \frac{\partial}{\partial s}\Gamma(\phi_s,r) $$
where we recognize that the Mellin transform of our Schwartz family, $\phi_s$ must obey a third order (shallow water wave) PDE. Here we explicitly calculate the factor which we can quickly verify as a solution.
$$\Gamma(\phi,r,s)=\int_{\mathbb R^\times \cap ~(0,1)} |x|^r~\phi_s(x)~{dx\over |x|}=2 \sqrt{\frac{s}{r}}K_1(2\sqrt{r s})$$
In the link above, it is stated at the end that "The main difficulty is then to erect a numerically verifiable “barrier” that ensures that as time increases, no zeroes of large imaginary part enter this region. This is done by use of the argument principle."
Comparing $\phi_0$ to $\phi_8$ (where $8$ is an arbitrary large positive $s$ value) what are the qualitative differences between the nontrivial zeros on the critical line? For $\phi_0$ can I correctly conclude that this is equivalent to the Riemann hypothesis?