Critical semilinear damped wave equations beyond moduli of continuity
Abstract
We consider the Cauchy problem for semilinear classical damped wave equations with general nonnegative nondecreasing nonlinearities, without imposing a modulus-of-continuity structure or convexity. A Dini-type integral condition expressed directly in terms of the nonlinearity gives small data global in-time existence and extends the nonlinear framework of Ebert--Girardi--Reissig [Math. Ann. 378 (2020), 1311--1326]. When this Dini-type integral diverges and the effective total mass of the initial data is positive, convergence of the Keller--Osserman tail integral is necessary and sufficient for finite-time blow-up of the maximal mild Sobolev solution, particularly, divergence of the tail integral gives global grow-up. The resulting integral criteria distinguish the contributions of the nonlinearity near zero and at infinity in governing small data global in-time existence with diffusion decay, global grow-up, and finite-time blow-up.
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BibTeX
@misc{chen2026critical,
title = {{Critical semilinear damped wave equations beyond moduli of continuity}},
author = {Wenhui Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27862},
eprint = {2609.27862},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27862},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Critical semilinear damped wave equations beyond moduli of continuity
AU - Wenhui Chen
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.27862
UR - https://arxiv.org/abs/2609.27862
AB - We consider the Cauchy problem for semilinear classical damped wave equations with general nonnegative nondecreasing nonlinearities, without imposing a modulus-of-continuity structure or convexity. A Dini-type integral condition expressed directly in terms of the nonlinearity gives small data global in-time existence and extends the nonlinear framework of Ebert--Girardi--Reissig [Math. Ann. 378 (2020), 1311--1326]. When this Dini-type integral diverges and the effective total mass of the initial data is positive, convergence of the Keller--Osserman tail integral is necessary and sufficient for finite-time blow-up of the maximal mild Sobolev solution, particularly, divergence of the tail integral gives global grow-up. The resulting integral criteria distinguish the contributions of the nonlinearity near zero and at infinity in governing small data global in-time existence with diffusion decay, global grow-up, and finite-time blow-up.
ER -