Sharp lifespan for the three-dimensional small-data fully nonlinear wave equation $\Box u+G(u',u'')=0$ (Hörmander, 1989)

For the Cauchy problem $\Box u+G(u',u'')=0$ in $\mathbb{R}^{1+3}$, where $\Box=\partial_t^2-\Delta$, $G\in C^\infty$ vanishes to second order at the origin, and the initial data are compactly supported of size $\varepsilon$, Hörmander (1989) proved the lifespan lower bound $\liminf_{\varepsilon\to0}\varepsilon\log T_\varepsilon\ge A$: the classical solution exists beyond times $e^{B/\varepsilon}$ for every $B<A$, where the constant $A$ is explicitly computable from the Friedlander radiation field of the data and the quadratic part of $G$ through a generalized Burgers profile equation, and the rescaled solution converges to that profile. This problem asks to prove the matching upper bound, hence $\lim_{\varepsilon\to0}\varepsilon\log T_\varepsilon=A$, for general non-symmetric data in three dimensions — sharp lifespan constants are known in two dimensions and in three dimensions only under radial symmetry for special model equations. The literature through September 2026 records no matching upper bound at this generality.

Background

The small-data problem and the lifespan. For a nonlinear wave equation in three space dimensions with initial data of size $\varepsilon>0$, the lifespan $T_\varepsilon$ is the supremum of the times up to which a classical ($C^\infty$) solution exists. The central question of the small-data theory is the sharp asymptotic behavior of $T_\varepsilon$ as $\varepsilon\to0$: whether $T_\varepsilon=\infty$ (global existence), whether $T_\varepsilon$ grows algebraically in $1/\varepsilon$, or whether $T_\varepsilon$ grows exponentially, $T_\varepsilon\approx e^{c/\varepsilon}$ (almost global existence). By the dependence of the nonlinearity one distinguishes semilinear equations (the nonlinearity involves $u$ and its first derivatives $u'$), quasilinear equations (the coefficients multiplying the second derivatives depend on $u$ and $u'$), and fully nonlinear equations (arbitrary smooth dependence on the second derivatives $u''$ as well); this problem concerns the fully nonlinear class $\Box u+G(u',u'')=0$, where $\Box=\partial_t^2-\Delta$, $G\in C^\infty$ vanishes to second order at the origin, and the data are $u(0,x)=\varepsilon u_0(x)$, $\partial_t u(0,x)=\varepsilon u_1(x)$ with $u_0,u_1\in C_0^\infty(\mathbb{R}^3)$.

Dimension three as the borderline; the null condition. The free wave in three spatial dimensions decays at the borderline rate $t^{-1}$, so quadratic nonlinearities sit exactly at the threshold between global existence and finite-time blowup. John (1979) showed that blowup does occur for nonlinear wave equations in three dimensions; John–Klainerman (1984) proved almost global existence, $T_\varepsilon\ge e^{c/\varepsilon}$, for general quadratic nonlinearities; and the null condition of Klainerman (1986) — the requirement that the quadratic part of the nonlinearity vanish on every null pair of directions (pairs of vectors with zero Minkowski inner product), which rules out the resonant interactions along the light cone responsible for blowup — identifies the special quadratic forms for which global existence holds in three dimensions. In dimensions $n\ge4$ the faster decay gives global small-data existence (Christodoulou 1986 and others, as recorded in the 2026 review table of Ding–Hou–Yin).

Hörmander's theorem (the source). Hörmander (1987, Lecture Notes in Mathematics 1256, quasilinear case; 1989, fully nonlinear case — the source of this problem, doi:10.1007/BF02585465) showed that the long-time behavior is governed by a generalized Burgers equation satisfied by the profile of the solution near the outgoing light cone. The Friedlander radiation field — the asymptotic coefficient of the free wave at future null infinity, which encodes each linear solution by a function on the sphere of outgoing directions together with the distance along the cone — is $F(\omega,q)=\frac{1}{4\pi}(R(\omega,q;u_1)-\partial_q R(\omega,q;u_0))$, where $R(\omega,q;g)=\int_{\langle\omega,y\rangle=q}g(y)\,dS(y)$ is the Radon transform (the integral of $g$ over the plane with normal $\omega\in S^2$ at signed distance $q$). The profile $U(\omega,s,q)$ solves, in the slow time $s=\varepsilon\log t$, the equation $\partial_s U_q=a(\omega)(\partial_q U_q)^2+2b(\omega)U_q\,\partial_q U_q+c(\omega)U_q^2$ with $U(\omega,0,q)=F(\omega,q)$, where $a,b,c$ are the three coefficients of the quadratic part of $G$ in the null frame (a basis containing two directions tangent to the scaled light cones and two null directions transverse to them) of the direction $\omega$. The profile is smooth exactly for $s<A$, where $1/A$ is the maximum over $(\omega,q)$ of $\sqrt{I_1(\omega,q)}+I_2(\omega,q)$ with $I_1=a(aF_{qqq}^2+2bF_{qqq}F_{qq}+cF_{qq}^2)$ and $I_2=aF_{qqq}+2bF_{qq}+cF_q$ (derivatives in $q$; non-real values of the square root omitted); some third derivatives of $U$ blow up as $s\uparrow A$ while $U$ and its first and second derivatives remain bounded. The main theorem of the 1989 paper: for every $B<A$ and sufficiently small $\varepsilon$ the classical solution exists on $0\le t\le e^{B/\varepsilon}$, and the rescaled solution $\varepsilon^{-1}r\,u_\varepsilon(t,r\omega)$ (with $t=e^{s/\varepsilon}$, $r=t+q$) converges uniformly to $U(\omega,s,q)$; in particular $\liminf_{\varepsilon\to0}\varepsilon\log T_\varepsilon\ge A$. The lower bound is proved through an explicit approximate solution whose residual is integrable over the exponential time scale. Thus $A$ is the conjecturally sharp lifespan exponent. For the reduced one-dimensional generalized Burgers equations the same paper derives complete, explicit, two-sided lifespan formulas in all parameter regimes (the regimes $a=0$ and $b^2-ac$ negative, zero, positive), including an exceptional class of global data; but at the level of the full three-dimensional Cauchy problem only the lower bound is known for general data.

Surrounding sharp results. In two space dimensions sharp lifespan laws are known for quadratic quasilinear equations: Alinhac (1999) obtained a complete asymptotic expansion of the lifespan and cusp-type blowup (the graph of the solution develops a cusp-shaped derivative singularity along a characteristic rather than a jump) for non-rotationally-invariant data, with later extensions to coefficients depending on $u$ and $\nabla u$ (Ding–Witt–Yin 2013) and to fully nonlinear equations (Han–Jin 2016); the 2026 research paper of Ding–Hou–Yin records the sharp two-dimensional law $\lim_{\varepsilon\to0+}\varepsilon\sqrt{T_\varepsilon^C}=\tilde\tau_0$, with $\tilde\tau_0=(\max Q_{1st}(\tilde\omega)\,\partial_\sigma^2\tilde F_0)^{-1}$ (their equation (1.22)). In three dimensions, sharp asymptotics are known only for special model equations under radial symmetry (Ding–Witt–Yin 2013); for general data with the null condition failing, the same paper records only the lower bound $T_\varepsilon\ge e^{C/\varepsilon}$. As background on the adjacent extended class $\Box u+G(u,u',u'')=0$, which is not part of this problem: Lindblad (1990) proved in $1+3$ dimensions that the lifespan exceeds $c\,\varepsilon^{-2}$ without the hypothesis $\partial_u^2 G(0)=0$ and that almost global existence $T_\varepsilon\ge e^{c/\varepsilon}$ holds for scalar equations with it; neither order is known to be sharp in general (Metcalfe–Rhoads 2023 for systems), and Hörmander's 1991 sequel notes that in low dimension the results are not always optimal when $G$ depends on $u$ itself. Likewise as background only: the modern continuation-past-blowup theory — shock formation for quasilinear waves whenever the null condition fails (Speck 2016) and the shock development problem for compressible Euler flow (Christodoulou 2019; Christodoulou–Lisibach 2016; Abbrescia–Speck 2022) — exists for quasilinear and first-order systems, not for the fully nonlinear class of this problem.

Problem Statement

Consider, in three space dimensions, the Cauchy problem

$$\Box u+G(u',u'')=0\ \text{ in }\ \mathbb{R}^{1+3},\qquad u(0,x)=\varepsilon u_0(x),\quad \partial_t u(0,x)=\varepsilon u_1(x),$$

where $\Box=\partial_t^2-\Delta$, $u'=(\partial_\alpha u)_{\alpha=0,1,2,3}$ denotes the first derivatives, $u''=(\partial_{\alpha\beta}u)_{0\le\alpha\le\beta\le3}$ the second derivatives, $G\in C^\infty$ vanishes together with its first derivatives at the origin, $u_0,u_1\in C_0^\infty(\mathbb{R}^3)$, and $0<\varepsilon\ll1$. Let $T_\varepsilon$ be the lifespan of the classical solution (the supremum of times up to which a $C^\infty$ solution exists). Let $a(\omega),b(\omega),c(\omega)$, $\omega\in S^2$, be the three coefficients of the quadratic part of $G$ in the null frame (a basis of two directions tangent to the scaled light cones and two null directions transverse to them) of the light-cone direction $\omega$, as defined in Hörmander (1989); let $F(\omega,q)=\frac{1}{4\pi}(R(\omega,q;u_1)-\partial_q R(\omega,q;u_0))$ be the Friedlander radiation field of the linear data, where $R(\omega,q;g)=\int_{\langle\omega,y\rangle=q}g(y)\,dS(y)$ is the Radon transform (the integral of $g$ over the plane with normal $\omega$ at signed distance $q$); and define

$$\frac{1}{A}=\max_{\omega\in S^2,\,q\in\mathbb{R}}\Big(\sqrt{a(\omega)\big(a(\omega)F_{qqq}^2+2b(\omega)F_{qqq}F_{qq}+c(\omega)F_{qq}^2\big)}+a(\omega)F_{qqq}+2b(\omega)F_{qq}+c(\omega)F_q\Big),$$

with derivatives taken in $q$ and the convention that non-real values of the square root are omitted; equivalently, $A$ is the maximal slow time $s=\varepsilon\log t$ up to which the generalized Burgers profile

$$\partial_s U_q=a(\omega)(\partial_q U_q)^2+2b(\omega)U_q\,\partial_q U_q+c(\omega)U_q^2,\qquad U(\omega,0,q)=F(\omega,q),$$

remains smooth (some third derivatives of $U$ blow up as $s\uparrow A$). Hörmander (1989) proved $\liminf_{\varepsilon\to0}\varepsilon\log T_\varepsilon\ge A$, and this remains the best known bound for general data.

The problem is to decide whether this lower bound is sharp. For data $u_0,u_1\in C_0^\infty(\mathbb{R}^3)$ with $0<A<\infty$ — that is, the displayed maximum is positive, equivalently the generalized Burgers profile ceases to be smooth at the finite slow time $A$ — and with no radial symmetry or other structural restriction on the data, prove or disprove that

$$\lim_{\varepsilon\to0}\varepsilon\log T_\varepsilon=A,$$

equivalently, establish the matching upper bound $\limsup_{\varepsilon\to0}\varepsilon\log T_\varepsilon\le A$. When the displayed maximum is $\le 0$ the profile is globally smooth and $A=\infty$ (the exceptional global-data regime of the reduced Burgers analysis), the displayed equation for $1/A$ applying only when this maximum is positive; such data lie outside the conjecture's scope.

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