Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations
Abstract
For the 3D semilinear Euler-Poisson-Darboux equation □ u+μ/t∂tu=|u|p, where t≥1, μ>0 and p>1, it is conjectured that there is a critical exponent pcrit(3,μ)=max{ps(3+μ), pf(3)} with the Strauss exponent ps(3+μ)=(μ+4+√(μ2+16μ+32))/(2(μ+2)) and the Fujita exponent pf(3)=5/3 such that when p>pcrit(3,μ), the small data solution u exists globally, otherwise, when 1<p≤ pcrit(3,μ), the solution u can blow up in finite time. It is pointed out that for 1<p≤ pcrit(3,μ), the blowup of solution u has been shown. However, it is still open for the global existence of small solution u when p>pcrit(3,μ). Note that pcrit(3,μ)=ps(3+μ) for 0<μ<14/5 and pcrit(3,μ)=pf(3) for μ≥14/5. In the recent paper [16], the authors have obtained the global small solution u for p>max{5/3,1+2/μ} and μ≥14/5. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution u in the remaining range of 5/3<p≤1+2/μ and 14/5≤μ<3. Therefore, for the radially symmetric case and μ≥14/5, the global existence problem of small data solution u is solved when p>pcrit(3,μ)=5/3.
Related
- Global existence for the semilinear regular Euler-Poisson-Darboux-Tricomi equation, II
- Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion
- Poisson problem with Hardy operator on lattice graph $ \mathbb{Z}^d$ and the application to semilinear equations
- Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
- Universal Hölder estimates for stable solutions to semilinear equations of fractional order
BibTeX
@misc{li2026global,
title = {{Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations}},
author = {Qianqian Li and Qiao Xin and Huicheng Yin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29258},
eprint = {2608.29258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29258},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations
AU - Qianqian Li
AU - Qiao Xin
AU - Huicheng Yin
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29258
UR - https://arxiv.org/abs/2608.29258
AB - For the 3D semilinear Euler-Poisson-Darboux equation □ u+μ/t∂_tu=|u|^p, where t≥1, μ>0 and p>1, it is conjectured that there is a critical exponent p_crit(3,μ)=max{p_s(3+μ), p_f(3)} with the Strauss exponent p_s(3+μ)=(μ+4+√(μ^2+16μ+32))/(2(μ+2)) and the Fujita exponent p_f(3)=5/3 such that when p>p_crit(3,μ), the small data solution u exists globally, otherwise, when 1<p≤ p_crit(3,μ), the solution u can blow up in finite time. It is pointed out that for 1<p≤ p_crit(3,μ), the blowup of solution u has been shown. However, it is still open for the global existence of small solution u when p>p_crit(3,μ). Note that p_crit(3,μ)=p_s(3+μ) for 0<μ<14/5 and p_crit(3,μ)=p_f(3) for μ≥14/5. In the recent paper [16], the authors have obtained the global small solution u for p>max{5/3,1+2/μ} and μ≥14/5. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution u in the remaining range of 5/3<p≤1+2/μ and 14/5≤μ<3. Therefore, for the radially symmetric case and μ≥14/5, the global existence problem of small data solution u is solved when p>p_crit(3,μ)=5/3.
ER -