Sharp global existence for nonradial derivative wave equations

Determine the sharp global-existence threshold for general nonradial initial data for Euclidean wave equations with derivative-type nonlinearities, including the validity of the Glassey critical exponent beyond the dimensions and symmetry classes already treated.

Background

The paper reviews the Glassey conjecture for Euclidean wave equations whose nonlinearities depend on time derivatives. Glassey conjectured the critical exponent to be p_G = 1 + 2/(n−1). Although the blow-up portion is known and the global-existence problem has been resolved in dimensions two and three, the sharp global-existence question for general nonradial data in higher dimensions remains unresolved.

The present paper does not attempt to identify the corresponding critical exponent for the positive-mass Klein–Gordon problem; instead, it proves a small-data global-existence and scattering result in a restricted exponent range under spherical symmetry and additional metric assumptions.

References

More specifically, in his review of Sideris' work , Glassey conjectured that the critical exponent should be $p_G = 1 + \frac{2}{n-1}$. A proof of the blow-up part of this conjecture was given by Zhou . Regarding global existence, the cases $n=2,3$ were treated independently by Hidano-Tsutaya and Tzvetkov . In higher dimensions, the sharp global-existence problem for general nonradial data remains open; radial-data results include, for example, Hidano-Wang-Yokoyama .

— Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications  (2610.11827 - Dai et al., 8 Oct 2026) in Section 1, subsection “Applications,” paragraph discussing the Glassey conjecture

For general, not necessarily Hamiltonian, Klein-Gordon equations with nonlinearities of the form $F(u,u_t,\nabla u)$, energy heuristics suggest the threshold $p>1+\frac{2}{n}$ in the Euclidean setting; see Lindblad-Sogge , Keel-Tao , and the references therein. The corresponding sharp theory for nonlinearities involving only derivatives has not yet been established in general.

— Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications  (2610.11827 - Dai et al., 8 Oct 2026) in Section 1, subsection “Applications,” paragraph following the discussion of the Glassey conjecture