The combined effect in one space dimension beyond the general theory for nonlinear wave equations

TL;DR

This paper investigates the unique combined effect of two nonlinear terms in one-dimensional semilinear wave equations, revealing phenomena only when the initial speed's integral is zero and improving lifespan estimates over general theories.

Contribution

It introduces new lifespan estimates for wave equations with combined nonlinear effects, surpassing existing general theory results in specific cases.

Findings

The combined effect occurs only when the initial speed integral is zero.

Lifespan estimates are improved compared to general theory.

The phenomenon is specific to one-dimensional semilinear wave equations.

Abstract

In this paper, we show the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Such a special phenomenon appears only in the case that the total integral of the initial speed is zero. It is remarkable that, including the combined effect case, our results on the lifespan estimates are partially better than those of the general theory for nonlinear wave equations.