Long time existence for semilinear wave equations with the inverse-square potential

TL;DR

This paper establishes long-time existence results for semilinear wave equations with inverse-square potential, identifying the critical exponent and lifespan bounds, and extends analysis to extreme potential cases.

Contribution

It introduces a novel approach by transforming the equation into a fractional dimensional wave equation and analyzes its fundamental solution, addressing cases previously unexplored.

Findings

Determined the critical exponent dividing global existence and blow-up.

Established sharp lower bounds of lifespan for solutions.

Extended analysis to extreme inverse-square potential cases.

Abstract

In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a "fractional dimensional" wave equation and analyzing the properties of its fundamental solution, we establish a long-time existence result, for sufficiently small, spherically symmetric initial data. Together with the previously known blow-up result, we determine the critical exponent which divides the global existence and finite time blow-up. Moreover, the sharp lower bounds of the lifespan are obtained, except for certain borderline case. In addition, our technology allows us to handle an extreme case for the potential, which has hardly been discussed in literature.