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RE: Original research: Modeling of reaction-diffusion transport into a core-shell geometry
That's a nice overview of your research manuscript.
I haven't looked in detail at the equations yet but on first glance it might be possible to rigorously prove the existence of solutions satisfying the boundary conditions (by analytical methods or rigorous numerics). Is this something you would be interested in?
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If you're interested, the problem I want to solve involves modifying this problem to better represent the coupling of glucose and oxygen.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3138450/
In part, you might think that this will hamper the cell life, because glucose is about 100X bigger than O2, but the consumption rate is coupled, so we have diminished consumption of O2. It might be that we have a system completely saturated with O2 so the cell life isn't limited by oxygen, but something else.
It is also noteworthy that I'm only looking at the steady state solution. If we are thinking about Insulin, the transient solution is also very important. If, for example, the cells get the signal to dump insulin, but it takes too long for the insulin to reach the blood, then there might be an overshoot and the patient will have a hypoglycemic response, which also can be deadly.
I think getting a transient solution is going to be a bear. I tried expressing the solution for the simple problem given here as a Fourier expansion, and the results wouldn't converge.
Again, I have zero funds for this, so as a result I find myself waiting to find an undergraduate engineering student that is willing to work for free. I'd try to write for financial support, but since this is way outside of my area of experience, I have almost to probability of funding. (Funding agents very much like to support you to continue the same work you already have proven yourself with.)
I think it is first best to make the results in the manuscript more rigorous and then see if it is possible to extend these to the coupled system.
A steady state solution is not useful if you don't have well-posedness and stability results for the underlying PDE. A full stability results is always complicated but a linear stability result might be possible. But for this you need to obtain the existence of the steady state solution. I wil discord dm you about it at a later time.