Self-similar solutions, regularity and time asymptotics for a nonlinear diffusion equation arising in game theory

TL;DR

This paper investigates the long-term behavior of a non-local nonlinear diffusion equation modeling the rock-paper-scissors game, characterizing self-similar solutions and proving convergence of solutions to these profiles over time.

Contribution

It provides a complete characterization of self-similar solutions and establishes convergence rates for the solutions of the nonlinear diffusion equation in a game-theoretic context.

Findings

Identification of self-similar solutions

Proof of convergence to self-similar profiles

Algebraic rate of convergence

Abstract

In this article, we study the long-time asymptotic properties of a non-linear and non-local equation of diffusive type which describes the rock-paper-scissors game in an interconnected population.We fully characterize the self-similar solution and then prove that the solution of the initial-boundary value problem converges to the self-similar profile with an algebraic rate.