Elliptic and Parabolic Equations - Abstract
Khrabustovskyi, Andrii
In this work we derive some qualitative properties of a complicated linear reaction-diffusion system in a simple domain homogenizing a simple diffusion equation on a complicated Riemannian manifold. Linear reaction-diffusion systems of parabolic type play an important role in applied mathematics. They describe, for instance, the transport of particles of various species in a medium and the transformation of the particles into each other. The transport can be forced by local (diffusion) and nonlocal interaction (jumps) of the particles with the medium, moreover, the transport and the reactions can be nonlocal in time (memory effects). We consider an initial boundary value problem for such a system.
We prove the existence of its solution and show that this solution preserves positivity (if the initial vector-function is positive) and converges exponentially to some constant as time goes to infinity (for the diffusion equation with Neumann boundary condition this constant is the mean value of the initial function). We give an explicit expression for this constant by proving a suitable Poincare inequality.
The proof of these results is based on the following trick. We construct a special Riemannian manifold depending on a small parameter in such a way that its topological genus increases as this parameter goes to zero. On this manifold we consider a simple diffusion equation. It turns out that the manifold can be chosen in such a way that solution of this equation converges (in some sense) to the solution of the original initial boundary value problem. Using this and the facts that the diffusion equation on manifolds preserves positivity and its solution converges to an easily calculated constant, we can obtain analogous properties for the original system. This method gives a microscopic interpretation of the terms of the system and allows us to “guess” easily the limit constant.
This work was done in collaboration with Dr. H. Stephan.