Stationary states of reaction-diffusion and Schr\"odinger systems with inhomogeneous or controlled diffusion

TL;DR

This paper classifies and analyzes positive stationary states in reaction-diffusion and Schrödinger systems with inhomogeneous or controlled diffusion, extending previous homogeneous diffusion results to more general media.

Contribution

It generalizes existing classification and nonexistence theorems from homogeneous to inhomogeneous and controlled diffusion systems, broadening their applicability.

Findings

Results valid for heterogeneous media

Extended classification to controlled diffusions

Confirmed nonexistence under certain conditions

Abstract

We obtain classification, solvability and nonexistence theorems for positive stationary states of reaction-diffusion and Schr\"odinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions. We show, with different proofs, that the results obtained in [ARMA, 213 (2014), 129-169] for models with homogeneous diffusion are valid for general heterogeneous media, and even for controlled inhomogeneous diffusions.