The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions

TL;DR

This paper investigates the lifespan of solutions to semilinear wave equations with scale-invariant damping in two dimensions, revealing a threshold effect that distinguishes wave-like and heat-like behaviors and providing new estimates for critical exponents.

Contribution

It extends lifespan analysis to two dimensions, identifying a unique damping constant threshold and deriving novel estimates for critical exponents in this setting.

Findings

Threshold damping constant separates wave-like and heat-like behaviors.

Lifespan estimates depend on the integral of initial data.

New estimates for critical exponent in two-dimensional case.

Abstract

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed. The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between "wave-like" domain and "heat-like" domain. As a result, we obtain a new type of estimate especially for the critical exponent.