A revisit via slicing method on a quadratic semilinear wave equation in two space dimensions

TL;DR

This paper presents a simplified iterative proof using slicing techniques to analyze blow-up behavior of solutions to a quadratic semilinear wave equation in two dimensions, with implications for numerical analysis.

Contribution

It introduces a straightforward proof method for lifespan estimates, improving understanding of blow-up phenomena in critical nonlinear wave equations.

Findings

Established a blow-up result for the equation in two dimensions.

Provided a simple iterative proof technique using slicing.

Connected the analysis to applications in numerical methods.

Abstract

In this paper, we are focusing on proofs of a blow-up result for a quadratic semilinear wave equation in two space dimensions. There is a logarithmic loss in estimating the lifespan of a classical solution if the 0th moment of the initial speed does not vanish. This result is already known with almost sharp constants. But in order to have a direct application to the numerical analysis, we show a simple proof by iteration argument for a point-wise estimate of the solution with a slicing technique. Such a research direction can be found in the case of critical nonlinearity.