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

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 weak form with the special choice of the test function.

But in order to have direct applications to the numerical analysis, we show the simple proof by iteration argument of point-wise estimates of the solution with the slicing technique.