On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics

TL;DR

This paper investigates a degenerate non-local parabolic PDE modeling infinite dimensional replicator dynamics, establishing existence of solutions, their long-term behavior, and conditions leading to finite-time blow-up.

Contribution

It introduces a new analysis of a degenerate, non-local PDE related to replicator dynamics, proving existence, convergence, and blow-up phenomena.

Findings

Solutions exist locally and are positive.

Small initial mass leads to solutions converging to zero.

Large initial mass causes finite-time blow-up with global blow-up set.

Abstract

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u \Delta u + u \int_\Omega |\nabla u|^2 \] in bounded domains $\Omega\subset\mathbb{R}^n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics