# Global existence of solutions of semilinear heat equation with nonlinear memory condition

## TL;DR

This paper investigates conditions under which solutions to a semilinear heat equation with nonlinear memory either exist globally or blow up in finite time, depending on the long-term behavior of variable coefficients.

## Contribution

It provides new criteria for global existence and blow-up of solutions for a semilinear heat equation with nonlinear memory boundary conditions, considering variable coefficient behavior.

## Key findings

- Global existence is guaranteed under certain coefficient conditions.
- Solutions can blow up in finite time depending on coefficient behavior.
- The results depend on the asymptotic behavior of variable coefficients as time approaches infinity.

## Abstract

We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as $t \to \infty.$