Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

TL;DR

This paper investigates blowup solutions in a nonlocal parabolic PDE related to the Gierer-Meinhardt system, revealing a novel blowup speed with a logarithmic correction under critical conditions.

Contribution

It introduces a new blowup speed phenomenon with a log correction term for a nonlocal PDE modeling the Gierer-Meinhardt system under critical parameters.

Findings

Identification of blowup solutions with log correction speed

New blowup behavior in critical regimes

Analysis of the nonlocal PDE's blowup dynamics

Abstract

We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in constructing blowup solutions under some critical regimes. We had a complete new phenomenon of the blowup speed with a log correction term.