The Threshold between Effective and Noneffective Damping for Semilinear Waves

TL;DR

This paper investigates the conditions under which small data solutions to semilinear wave equations with scale-invariant damping exist globally, analyzing decay rates and extending results to models with polynomial and exponential propagation speeds.

Contribution

It provides new estimates for solutions and energy decay rates, and extends analysis to models with different propagation speeds, clarifying the damping effectiveness threshold.

Findings

Established decay estimates matching linear problem rates

Extended results to polynomial and exponential propagation models

Identified the damping threshold for solution effectiveness

Abstract

In this paper we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with scale-invariant damping. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. We extend our results to a model with polynomial speed of propagation and to a model with an exponential speed of propagation.