# Critical regularity of nonlinearities in semilinear classical damped wave equations

## TL;DR

This paper investigates the critical regularity conditions on nonlinearities in semilinear damped wave equations, establishing sharp thresholds for global existence versus blow-up of small solutions based on the modulus of continuity.

## Contribution

It provides precise criteria on the modulus of continuity for nonlinearities to distinguish between global solutions and blow-up in semilinear damped wave equations.

## Key findings

- Sharp conditions on the modulus of continuity for global existence.
- Thresholds between stability and blow-up for small data solutions.
- Characterization of nonlinearities leading to different solution behaviors.

## Abstract

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is a modulus of continuity. Our goal is to obtain sharp conditions on $\mu$ to obtain a threshold between global (in time) existence of small data solutions (stability of the zerosolution) and blow-up behavior even of small data solutions.