N-barrier maximum principle for degenerate elliptic systems and its application

TL;DR

This paper introduces the N-barrier maximum principle for degenerate elliptic systems, extending previous results to nonlinear diffusion equations and providing bounds for solutions, with applications in ecology.

Contribution

It develops the N-barrier maximum principle for nonlinear degenerate elliptic systems, including the porous medium type, and applies it to ecological coexistence problems.

Findings

Established a priori bounds for solutions to nonlinear degenerate elliptic systems.

Extended maximum principle results from linear to nonlinear diffusion equations.

Proved nonexistence of waves in a three-species ecological model.

Abstract

In this paper, we prove the N-barrier maximum principle, which extends the result in [5] from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. As an application of the N-barrier maximum principle to a coexistence problem in ecology, we show the nonexistence of waves in a three-species degenerate elliptic systems.