Influence of variable coefficients on global existence of solutions of semilinear heat equations with nonlinear boundary conditions

TL;DR

This paper investigates how variable coefficients influence whether solutions to certain semilinear heat equations with nonlinear boundary conditions exist globally or blow up in finite time, depending on their long-term behavior.

Contribution

It provides new conditions linking the asymptotic behavior of variable coefficients to the global existence or blow-up of solutions.

Findings

Conditions for global existence based on coefficient behavior as t→∞

Criteria for finite-time blow-up of solutions

Dependence of solution behavior on variable coefficients

Abstract

We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t \to \infty.$