The blow-up theorem of a discrete semilinear wave equation

TL;DR

This paper proves that a discretized nonlinear wave equation exhibits blow-up solutions under conditions similar to the continuous case, mirroring the blow-up behavior of the original PDE.

Contribution

It establishes a rigorous connection between blow-up solutions of the continuous nonlinear wave equation and its discrete counterpart.

Findings

Discretized solutions mirror blow-up behavior of the continuous equation

Blow-up conditions are preserved under discretization

Theoretical proof of blow-up in the difference equation

Abstract

In this paper, the discretization of a nonlinear wave equation whose nonlinear term is a power function is introduced. The difference equation derived by discretizing the nonlinear wave equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial value 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 value problem for the introduced difference equation, the introduced difference equation has blow-up solutions having characteristics corresponding to the characteristics of the blow-up solutions for the original equation.