An Exact Mass Threshold for Global Existence and Blow-up in Semilinear Anti-de Sitter Equations

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 $${\phi }_{tt}-e^{2t}\mathrm{\Delta }\phi +n{\phi }_t+m^2\phi =\mathrm{\mid }\phi {\mathrm{\mid }}^p\text{in }[0,\mathrm{\infty })\times {\mathbb{R}}^d\mathrm{.}$$ 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 $m^2>-\frac{d(2n+d)}{4}\text{and}p>1.$ 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 $-d(n+d)<m^2\le -\frac{d(2n+d)}{4}\text{for every }p>1.$ In the complementary regime $m^2\le -d(n+d)$, 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.