Generalized Strauss conjecture for semilinear wave equations on $\mathbb{R}^3$

TL;DR

This paper investigates the critical conditions on the nonlinearity modulus for semilinear wave equations in three dimensions, establishing near-optimal thresholds that determine global existence versus blow-up, and disproving a recent conjecture.

Contribution

It provides almost sharp conditions on the modulus of continuity for the Strauss critical exponent in 3D wave equations, advancing understanding of nonlinear wave behavior.

Findings

Identifies near-sharp thresholds for global existence and blow-up.

Disproves the conjecture proposed in Chen 2024.

Establishes new criteria for the nonlinearity modulus .

Abstract

In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation $$\partial_{t}^2 u-\Delta_{\mathbb{R}^3}u=|u|^{p_S}\mu(|u|)\ ,u(0,x)=\varepsilon u_0,\ u_t(0,x)=\varepsilon u_1\ ,$$ where $p_S=1+\sqrt{2}$ is the Strauss critical index in $n=3$, and $\mu$ is a modulus of continuity. Inspired by Chen, Reissig\cite{Chen_2024} and Ebert, Girardi, Reissig\cite{MR4163528}, we investigate the sharp condition of $\mu$ as the threshold between the global existence and blow up with small data. We obtain the almost sharp results in this paper, which in particular disproves the conjecture in \cite{Chen_2024}.