Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition

TL;DR

This paper investigates the blow-up behavior of solutions to a semilinear heat equation with nonlinear nonlocal Neumann boundary conditions, establishing conditions for global existence, finite-time blow-up, and boundary-only blow-up.

Contribution

It provides new criteria for blow-up versus global existence and demonstrates boundary-only blow-up under specific conditions.

Findings

Criteria for finite-time blow-up versus global solutions

Conditions under which blow-up occurs only on the boundary

Global existence results for the semilinear heat equation

Abstract

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.