A revisit on the critical blow-up for semilinear wave equations in low space dimensions with slicing method

TL;DR

This paper reviews methods for estimating the lifespan of solutions to semilinear wave equations with critical exponents in low dimensions, emphasizing a new simple iterative slicing technique for point-wise estimates.

Contribution

It introduces a straightforward iterative slicing method for point-wise estimates, improving understanding of blow-up phenomena in low-dimensional wave equations.

Findings

The slicing method provides a clear proof of lifespan estimates.

Comparison with other methods highlights the simplicity of the proposed approach.

Results contribute to the theoretical understanding of critical blow-up in wave equations.

Abstract

In this reviewing paper, we are interested in the proof of estimating the lifespan of classical solutions of semilinear wave equations with the critical exponent from above especially in low space dimensions. There are a few ways to show the result by comparison argument with ODE via point-wise estimates, or by functional method via a weak form with a special choice of test functions. But in order to make a good much with the numerical analysis, we show a simple proof by iteration argument of a point-wise estimate of the solution with a slicing technique.