On the asymptotic profile of solutions to semilinear damped wave equations with critical nonlinearities

TL;DR

The paper analyzes the Cauchy problem for a semilinear damped wave equation with critical nonlinearity $\mathcal{N}(u)=|u|^{1+\frac{2}{n}}\mu(|u|)$, where $\mu$ is a modulus of continuity. It proves global existence for $1\le n\le 4$ under the Dini condition on $\mu$ and a mild growth control, establishing precise decay rates and Sobolev regularity for small data solutions. It further shows that, for large times, global solutions exhibit diffusion-dominated behavior and are asymptotically governed by the Gauss kernel, with the asymptotic mass $M$ incorporating both initial data and the nonlinear term. In the blow-up regime (non-Dini), the paper derives sharp lifespan estimates, showing $\Psi(T_{\varepsilon})$ scales like $\varepsilon^{-2/n}$, and extends lower-bound results to $n=4$, thereby completing the picture across dimensions up to four. Overall, the work extends known low-dimensional μ-conditions to $n=4$, clarifies the asymptotic profile of global solutions, and provides precise lifespan thresholds in the critical regime.

Abstract

In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} μ(|u|)$, where $μ$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020)) and Girardi (Nonlinear Differ. Equ. Appl. 32 (2025)), the authors obtained a sharp critical condition on $μ$ in low space dimensions $n=1,2,3$, which determines the threshold between global (in time) existence of small data solutions and blow-up of solutions in finite time. Our new results are to prove that this condition remains valid in dimension $n=4$, together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at $t \to \infty$ is identified by the Gauss kernel. Finally, a sharp lifespan estimate for local solutions is also derived in the case when blow-up occurs.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.1: Global existence
• Remark 1.1
• Remark 1.2
• Theorem 1.2: Asymptotic profiles
• Remark 1.3
• Lemma 2.1: Linear Estimates
• proof
• Lemma 2.2
• proof