Deng (2018) proved that every small-data solution of the general quadratic quasilinear multispeed Klein–Gordon system in three space dimensions is global and scatters to a linear solution, with no restrictions on the speeds cαc_\alpha and masses bαb_\alpha. In two space dimensions, where linear solutions decay only like t−1t^{-1}, the counterpart theory is incomplete: Deng's paper expects that for generic choices of speeds and masses — those satisfying the correct non-degeneracy and separation conditions, of which the mass non-resonance condition bα+bβ≠bγb_\alpha+b_\beta\neq b_\gamma is one — small-data global existence and scattering should still hold, while carefully chosen resonant systems violating it may admit small-data blow-up. Later work has settled parts of this: Zhu (2025) proved global existence and scattering in 2D for semilinear systems under two nondegeneracy conditions, and an unrefereed 2024 preprint constructs a small-data blow-up example indicating that the mass condition is necessary for generic nonlinearities. The open problem is to complete the two-dimensional theory for the quasilinear system: extend global existence and scattering beyond the semilinear class, determine whether the speed–mass alignment condition can be removed (or is itself necessary), and put the blow-up side on rigorous, self-contained footing.

Background

A single Klein–Gordon equation (∂t2−Δ+b2)u=N(u,∂u)(\partial_t^2-\Delta+b^2)u=N(u,\partial u) describes a massive field: plane waves obey the dispersion relation ω(ξ)=∣ξ∣2+b2\omega(\xi)=\sqrt{|\xi|^2+b^2}, and localized solutions spread and decay pointwise like t−n/2t^{-n/2} in nn space dimensions. For smooth, localized initial data of size ε≪1\varepsilon\ll1, the basic question is whether the solution exists globally in time and scatters — that is, converges as t→±∞t\to\pm\infty, in a natural Sobolev norm, to a solution of the underlying linear equation — or whether the nonlinearity can focus the dispersing waves and destroy the solution. In three dimensions the decay t−3/2t^{-3/2} is integrable in time, and this was settled for a single equation with quadratic nonlinearities independently by Klainerman and by Shatah (1985). In two dimensions the decay is only t−1t^{-1}, which is borderline: quadratic interactions are long-range, and small-data global existence becomes a delicate question that can genuinely fail.

Systems of coupled Klein–Gordon equations arise, for example, when the Euler–Poisson or Euler–Maxwell systems of plasma physics are reduced to their dispersive components. In a multispeed system each component has its own speed cαc_\alpha and mass bαb_\alpha: (∂t2−cα2Δ+bα2)uα=Qα(u,∂u,∂2u),1≤α≤d,(\partial_t^2-c_\alpha^2\Delta+b_\alpha^2)u_\alpha=Q_\alpha(u,\partial u,\partial^2 u),\qquad 1\le\alpha\le d, with quadratic quasilinear right-hand sides (nonlinearities involving second derivatives of the unknowns, such as u ∂2uu\,\partial^2u and ∂u ∂2u\partial u\,\partial^2u, plus purely semilinear quadratic terms depending only on uu and ∂u\partial u). The relative sizes of the speeds and masses govern the resonances of the system, and the two-dimensional small-data question for this general system is the subject of this problem.

The modern approach is the space-time resonance method of Germain, Masmoudi and Shatah. One rewrites the system in terms of profiles fσ=eitΛσvσf_\sigma=e^{it\Lambda_\sigma}v_\sigma (the linear flow pulled back), where Λσ(ξ)=cσ2∣ξ∣2+bσ2\Lambda_\sigma(\xi)=\sqrt{c_\sigma^2|\xi|^2+b_\sigma^2} is the dispersion relation; the Duhamel formula (the variation-of-constants integral rewriting of the equation) for the profile is an oscillatory integral whose phase is the difference of three dispersion relations, Φσμν(ξ,η)=Λσ(ξ)−Λμ(ξ−η)−Λν(η)\Phi_{\sigma\mu\nu}(\xi,\eta)=\Lambda_\sigma(\xi)-\Lambda_\mu(\xi-\eta)-\Lambda_\nu(\eta). Contributions concentrate near the space-time resonant set R={(ξ,η):Φ=∇ηΦ=0}R=\{(\xi,\eta):\Phi=\nabla_\eta\Phi=0\}. Control of the profile is measured in a Z-norm — a norm capturing both frequency and physical-space localization, whose choice is the crux of any proof: strong enough to imply integrable L∞L^\infty decay for the linear flow, weak enough to be propagated by the bilinear oscillatory integrals. The method is complemented by rotation vector fields Ωij=xi∂j−xj∂i\Omega_{ij}=x_i\partial_j-x_j\partial_i (from Klainerman's vector-field method), which exploit the asymptotic spherical symmetry of Klein–Gordon waves and buy angular localization in the bilinear estimates.

The landmark results for systems are: Hayashi, Naumkin and Wibowo (2008), many equations in 3D with equal speeds; Delort, Fang and Xue (2004), two equations in 2D with equal speed under a non-resonance condition; Germain (2011), two equations in 3D with different speeds and equal mass; Ionescu and Pausader (2014), the general quasilinear system in 3D under two nondegeneracy conditions; and Deng (2018), the general quasilinear system in 3D with all conditions removed. The two conditions of Ionescu–Pausader, in the two-dimensional notation of Zhu, are: (C1) the mass non-resonance bσ−bμ−bν≠0b_\sigma-b_\mu-b_\nu\neq0 for all σ,μ,ν\sigma,\mu,\nu, which excludes a degenerate resonance at zero frequency (the point (0,0)(0,0) entering the resonant set with vanishing phase derivatives to high order); and (C2) the speed–mass alignment (cμ−cν)(cμ2bν−cν2bμ)≥0(c_\mu-c_\nu)(c_\mu^2b_\nu-c_\nu^2b_\mu)\ge0 — the algebraic form used in the two-dimensional literature (Zhu's condition (1.2)) — a convenient algebraic condition implying the Hessian ∇η2Φ\nabla^2_\eta\Phi is nondegenerate on the resonant set away from the origin, which (C1) excludes. Deng removed both in 3D by combining rotation vector fields with a refined, log-loss analysis of the degenerate low-frequency regime.

For equal speeds the two-dimensional theory is in reasonably complete shape: Delort–Fang–Xue (quasilinear, non-resonant masses, localized data), Schottdorf (semilinear, non-resonant masses including equal masses, without decay assumptions on the data), and Katayama, Ozawa and Sunagawa, who showed that even when the masses satisfy the resonance relation, a null condition — a structural cancellation assumption on the nonlinearity in the sense of Delort–Fang–Xue — still yields global asymptotically free solutions. The genuinely open two-dimensional territory is the multispeed quasilinear system. There every difficulty sharpens: the t−1t^{-1} decay is not integrable, so energy estimates need TT∗TT^*-style arguments (estimates for the Duhamel operator composed with its adjoint) instead of a Gronwall scheme; iterated resonances (resonant outputs re-entering as resonant inputs) occur even under (C1) and (C2); and the rotation-vector-field gains are powerless against low-frequency cascades built from rescaled Schwartz functions (Schwartz functions, that is smooth functions all of whose derivatives decrease rapidly, dilated to concentrate near zero frequency) — the very mechanism expected to produce blow-up when the mass resonance bα+bβ=bγb_\alpha+b_\beta=b_\gamma is present. Deng, Ionescu and Pausader (2017) had earlier proved the two-dimensional Euler–Maxwell system for electrons global and scattering, a physically structured multispeed system that exhibits the behavior expected generically in 2D; the general system, however, remained open. Deng's preprint (§1.2.6) observes that the two-dimensional study is much harder: with carefully chosen speeds and masses one can construct a Klein–Gordon system that is expected to blow up in finite time even for small data, on a heuristic level, while for generic choices of speeds and masses — those with the correct non-degeneracy and separation conditions — one can still expect global well-posedness and scattering. The precise remaining questions are those listed by Zhu: whether the speed–mass condition (C2) can be removed (or a counterexample shows that (C2) is also necessary), and whether the semilinear theory extends to the quasilinear class with the symmetry assumption.

Problem Statement

Consider, on Rt×Rx2\mathbb{R}_t\times\mathbb{R}_x^2, the quadratic quasilinear multispeed Klein–Gordon system (∂t2−cα2Δ+bα2) uα=Qα(u,∂u,∂2u),1≤α≤d,(\partial_t^2-c_\alpha^2\Delta+b_\alpha^2)\,u_\alpha=Q_\alpha(u,\partial u,\partial^2 u),\qquad 1\le\alpha\le d, where the speeds cα>0c_\alpha>0 and masses bα>0b_\alpha>0 are arbitrary, and each QαQ_\alpha is a quadratic quasilinear nonlinearity Qα=∑β,γ∑j,kAαβγjk uγ ∂j∂kuβ+∑β,γ∑j,k,lBαβγjkl ∂luγ ∂j∂kuβ+Qα′(u,∂u),Q_\alpha=\sum_{\beta,\gamma}\sum_{j,k}A_{\alpha\beta\gamma}^{jk}\,u_\gamma\,\partial_j\partial_k u_\beta+\sum_{\beta,\gamma}\sum_{j,k,l}B_{\alpha\beta\gamma}^{jkl}\,\partial_l u_\gamma\,\partial_j\partial_k u_\beta+Q'_\alpha(u,\partial u), with AA, BB constant tensors symmetric in (α,β)(\alpha,\beta) (the symmetry assumption, needed for the symmetrization behind even the local theory) and Qα′Q'_\alpha an arbitrary constant-coefficient quadratic form in (u,∂u)(u,\partial u). Let (C1) denote the mass non-resonance condition bα+bβ≠bγb_\alpha+b_\beta\neq b_\gamma for all α,β,γ\alpha,\beta,\gamma (equivalently bσ−bμ−bν≠0b_\sigma-b_\mu-b_\nu\neq0 for all σ,μ,ν\sigma,\mu,\nu, the form used in the later two-dimensional literature), and (C2) the speed–mass alignment condition (cμ−cν)(cμ2bν−cν2bμ)≥0(c_\mu-c_\nu)(c_\mu^2b_\nu-c_\nu^2b_\mu)\ge0 for all μ,ν\mu,\nu. The problem is to determine the small-data behavior of this system in two space dimensions — the two-dimensional analogue of Deng's three-dimensional theorem — as a dichotomy in the speed and mass parameters.

(A) Sufficiency — prove or refute. Prove that, under (C1) alone, sufficiently small initial data (smooth and localized, small in the natural Sobolev, vector-field-weighted, profile/Z-type topology) lead to a unique global solution that scatters to a solution of the underlying linear system. This subsumes two independently refutable sub-claims: (i) the extension from the semilinear class (where it is known under (C1) and (C2), by Zhu) to the quasilinear class; and (ii) the removal of (C2). Alternatively, refute this expectation by exhibiting a system satisfying (C1) for which suitable small data produce finite-time blow-up or unbounded growth of a natural Sobolev-type norm; a counterexample that also satisfies (C2) refutes sub-claim (i), whereas one that satisfies (C1) but violates (C2) refutes sub-claim (ii) and shows that (C2) cannot be dropped.

(B) Necessity — establish rigorously that the mass non-resonance condition cannot be dropped for generic nonlinearities: construct a two-dimensional multispeed Klein–Gordon system with a degenerate mass resonance bα+bβ=bγb_\alpha+b_\beta=b_\gamma (and nonlinearity not enjoying a null-type cancellation, since null conditions are known to rescue the resonant equal-speed regime) and small data whose solution blows up in finite time or exhibits unbounded norm growth. The only construction currently available is an unrefereed preprint for one specific six-component system.

Parts (A) and (B) are the two sides of one objective: a small-data dichotomy for the two-dimensional system, identifying exactly which speed–mass configurations force global existence and scattering, and which permit small-data blow-up. A proof of (A) establishes the sufficiency of (C1) for every configuration satisfying it, whereas a refutation of (A) shows that (C1) alone does not suffice, without classifying the configurations for which it does; and the necessity of (C1) for generic nonlinearities is the content of (B).

References

  1. Yu Deng, Multispeed Klein–Gordon Systems in Dimension Three, International Mathematics Research Notices 2018(19), 6070–6144 (published online 2017), DOI 10.1093/imrn/rnx038; preprint arXiv:1602.01820, https://arxiv.org/abs/1602.01820
  2. Xilu Zhu, Global solutions of multispeed semilinear Klein-Gordon systems in space dimension two, Journal of Differential Equations 437 (2025), article 113279, DOI 10.1016/j.jde.2025.113279; preprint arXiv:2312.09596 (2023), https://arxiv.org/abs/2312.09596
  3. Xilu Zhu, A Blowup Solution of Multispeed Klein-Gordon System in Space Dimension Two with Small Initial Data, preprint arXiv:2402.03727 (2024, unpublished), https://arxiv.org/abs/2402.03727
  4. A. Ionescu and B. Pausader, Global solutions of quasilinear systems of Klein–Gordon equations in 3D, Journal of the European Mathematical Society 16 (2014), 2355–2431, DOI 10.4171/jems/489, https://doi.org/10.4171/jems/489
  5. J.-M. Delort, D. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions, Journal of Functional Analysis 211 (2004), 288–323, DOI 10.1016/j.jfa.2004.01.008, https://doi.org/10.1016/j.jfa.2004.01.008
  6. P. Germain, Global existence for coupled Klein-Gordon equations with different speeds, Annales de l'Institut Fourier 61 (2011), 2463–2506, DOI 10.5802/aif.2680, https://doi.org/10.5802/aif.2680
  7. Yu Deng, A. Ionescu and B. Pausader, The Euler–Maxwell System for Electrons: Global Solutions in 2D, Archive for Rational Mechanics and Analysis 225 (2017), 771–871, DOI 10.1007/s00205-017-1114-3, https://doi.org/10.1007/s00205-017-1114-3
  8. Nakao Hayashi, Pavel I. Naumkin and Ratno Bagus Edy Wibowo, Nonlinear scattering for a system of nonlinear Klein–Gordon equations, Journal of Mathematical Physics 49 (2008), 103501, DOI 10.1063/1.2990493, https://doi.org/10.1063/1.2990493
  9. Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Communications on Pure and Applied Mathematics 38 (1985), 631–641, DOI 10.1002/cpa.3160380512, https://doi.org/10.1002/cpa.3160380512
  10. Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Communications on Pure and Applied Mathematics 38 (1985), 685–696, DOI 10.1002/cpa.3160380516, https://doi.org/10.1002/cpa.3160380516
  11. Soichiro Katayama, Tohru Ozawa and Hideaki Sunagawa, A note on the null condition for quadratic nonlinear Klein–Gordon systems in two space dimensions, Communications on Pure and Applied Mathematics 65(9) (2012), 1285–1302, DOI 10.1002/cpa.21392, https://doi.org/10.1002/cpa.21392
  12. Tobias Schottdorf, Global existence without decay for quadratic Klein-Gordon equations, preprint arXiv:1209.1518 (2012), https://arxiv.org/abs/1209.1518
  13. Qian Zhang, Global solutions to quasilinear wave-Klein-Gordon systems in two space dimensions, Discrete and Continuous Dynamical Systems 45(7) (2025), 2317–2348, DOI 10.3934/dcds.2024167; preprint arXiv:2312.03287 (2023), https://arxiv.org/abs/2312.03287
  14. Istvan Kadar, On global behavior of classical effective field theories, Journal of Hyperbolic Differential Equations 21(1) (2024), 85–127, DOI 10.1142/S0219891624500036, https://doi.org/10.1142/S0219891624500036; preprint arXiv:2208.09194 (2022), https://arxiv.org/abs/2208.09194
  15. P. Germain, N. Masmoudi and J. Shatah, Global Solutions for 3D Quadratic Schrodinger Equations, International Mathematics Research Notices 2009(3) (2009), 414–432, DOI 10.1093/imrn/rnn135, https://doi.org/10.1093/imrn/rnn135