Blow-up of solutions to a Dirichlet problem for the discrete semi-linear heat equation

TL;DR

This paper investigates the blow-up behavior of solutions to a discretized semi-linear heat equation with Dirichlet boundary conditions, showing that the discrete model replicates the blow-up characteristics of the continuous original under similar conditions.

Contribution

It proves that the discretized difference equation exhibits blow-up solutions with properties analogous to those of the original semi-linear heat equation when certain conditions are satisfied.

Findings

Discrete solutions show blow-up behavior similar to the continuous case.

Blow-up characteristics are preserved under discretization.

Conditions for blow-up in the difference equation mirror those of the original equation.

Abstract

In this paper, the initial and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial and boundary problem for the original equation has blow-up solutions when a certain condition is met. We prove that when a similar condition as that of the original solution is met in the initial and boundary problem for the difference equation, the difference equation has blow-up solutions having characteristics corresponding to the characteristics of the blow-up solutions for the original equation.