An Exact Mass Threshold for Global Existence and Blow-up in Semilinear Anti-de Sitter Equations
Abstract
We study the influence of the mass parameter on global existence and finite-time blow-up for the semilinear anti-de Sitter Klein--Gordon equation ϕtt−e2tΔϕ+nϕt+m2ϕ=|ϕ|p in [0,∞)×Rd. After removing the damping term, we establish L2 energy estimates for the resulting Klein--Gordon equation with constant coefficient in the potential. They imply global existence for arbitrary sufficiently small energy data, without spatial localization, whenever m2>−(d(2n+d))/4 and p>1. In the remaining mass parameter interval above the structural threshold, an exact change to conformal time reduces the problem to a scale-invariant effective damped wave equation. Exponentially weighted energy estimates then extend global existence to −d(n+d)<m2≤−(d(2n+d))/4 for every p>1. In the complementary regime m2≤−d(n+d), we prove finite-time blow-up for every p>1 and for weak solutions with finite speed of propagation arising from nonnegative, nontrivial initial data. We further obtain lifespan estimates that are polynomial at the threshold and logarithmic below it. Thus, m2=−d(n+d) is the exact structural mass parameter threshold separating the global-existence and blow-up regimes.
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BibTeX
@misc{nascimento2026exact,
title = {{An Exact Mass Threshold for Global Existence and Blow-up in Semilinear Anti-de Sitter Equations}},
author = {Wanderley Nunes do Nascimento and Michael Reissig},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25239},
eprint = {2610.25239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25239},
}
RIS
TY - UNPB
TI - An Exact Mass Threshold for Global Existence and Blow-up in Semilinear Anti-de Sitter Equations
AU - Wanderley Nunes do Nascimento
AU - Michael Reissig
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25239
UR - https://arxiv.org/abs/2610.25239
AB - We study the influence of the mass parameter on global existence and finite-time blow-up for the semilinear anti-de Sitter Klein--Gordon equation ϕ_tt−e^2tΔϕ+nϕ_t+m^2ϕ=|ϕ|^p in [0,∞)×R^d. After removing the damping term, we establish L^2 energy estimates for the resulting Klein--Gordon equation with constant coefficient in the potential. They imply global existence for arbitrary sufficiently small energy data, without spatial localization, whenever m^2>−(d(2n+d))/4 and p>1. In the remaining mass parameter interval above the structural threshold, an exact change to conformal time reduces the problem to a scale-invariant effective damped wave equation. Exponentially weighted energy estimates then extend global existence to −d(n+d)<m^2≤−(d(2n+d))/4 for every p>1. In the complementary regime m^2≤−d(n+d), we prove finite-time blow-up for every p>1 and for weak solutions with finite speed of propagation arising from nonnegative, nontrivial initial data. We further obtain lifespan estimates that are polynomial at the threshold and logarithmic below it. Thus, m^2=−d(n+d) is the exact structural mass parameter threshold separating the global-existence and blow-up regimes.
ER -