Threshold and strong threshold solutions of a semilinear parabolic equation

TL;DR

This paper investigates the large-time behavior of radial positive solutions to a semilinear parabolic equation at the critical threshold between global existence and blow-up, focusing on threshold solutions.

Contribution

It analyzes the threshold and strong threshold solutions of a semilinear parabolic equation, providing insights into their large-time dynamics at the critical exponent.

Findings

Identification of conditions for global solutions and blow-up

Characterization of threshold solutions' asymptotic behavior

Insights into the borderline case between existence and blow-up

Abstract

If $p>1+2/n$ then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.