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Critical Curve for Three-Dimensional Radial Wave Systems with Distinct Speeds
Abstract
We study radially symmetric solutions to the three-dimensional semilinear wave system with distinct propagation speeds utt−c12Δu=|vt|p , vtt−c22Δv=|ut|q , c1, c2>0 , c1≠ c2 . For 1<p,q<2, we prove that Γ(p,q) = 2+max{1/p,1/q}−pq=0 is the exact critical curve separating small-data global existence from finite-time blow-up. More precisely, sufficiently small, compactly supported radial initial data give rise to global solutions when Γ(p,q)<0. On the other hand, for a class of initial data, the corresponding solutions blow up in finite time, with an upper bound for the lifespan, when Γ(p,q)≥ 0.
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BibTeX
@misc{dai2026critical,
title = {{Critical Curve for Three-Dimensional Radial Wave Systems with Distinct Speeds}},
author = {Wei Dai and Mengyun Liu and Chengbo Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07710},
eprint = {2611.07710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07710},
}
RIS
TY - UNPB
TI - Critical Curve for Three-Dimensional Radial Wave Systems with Distinct Speeds
AU - Wei Dai
AU - Mengyun Liu
AU - Chengbo Wang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07710
UR - https://arxiv.org/abs/2611.07710
AB - We study radially symmetric solutions to the three-dimensional semilinear wave system with distinct propagation speeds u_tt−c_1^2Δu=|v_t|^p , v_tt−c_2^2Δv=|u_t|^q , c_1, c_2>0 , c_1≠ c_2 . For 1<p,q<2, we prove that Γ(p,q) = 2+max{1/p,1/q}−pq=0 is the exact critical curve separating small-data global existence from finite-time blow-up. More precisely, sufficiently small, compactly supported radial initial data give rise to global solutions when Γ(p,q)<0. On the other hand, for a class of initial data, the corresponding solutions blow up in finite time, with an upper bound for the lifespan, when Γ(p,q)≥ 0.
ER -