发表机构
Ecole Polytechnique; Princeton University（巴黎综合理工学院; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文针对实值散焦能量超临界五次波动方程，构造了后向光锥中的离散自相似解，并通过计算机辅助证明其有限时间爆破，首次给出此类方程的实值爆破解。
AI 中文摘要
我们考虑定义在$\mathbb{R}^{1+10}$上的实值函数的散焦五次波动方程$\partial_t^2u-\Delta u+u^5=0$，该方程是能量超临界的。我们在后向光锥中构造了一个离散自相似解，形式为$$u(t,x)=(T-t)^{-1/2}W^*\big(\omega^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$其中非零轮廓$W^*$在其第一个变量上是$2\pi$-周期的，并且在光锥上及跨越光锥都是实解析的。截断其初始数据得到光滑、紧支撑、径向的数据，其解在后向光锥中与离散自相似解一致，在$[0,T]\times\mathbb{R}^{10}$上除锥顶点$(T,0)$外是光滑的，并在该处以自相似速率爆破。我们在此构造的解似乎是散焦能量超临界NLW的首个实值爆破解。实轮廓的存在性通过计算机辅助的Newton--Kantorovich论证在Fourier--Chebyshev系数的Banach代数中证明，并具有严格的误差控制。我们通过数值延拓复值方程的旋转自相似轮廓来找到近似轮廓。此步骤对爆破证明并非必要，因此作为非严格数值处理。主要的解析成分包括对线性部分模式算子的精确描述，这些算子在Chebyshev基下是超几何且上三角的，以及对其逆算子在高频处一致有界的估计。所有计算机辅助证明，包括所需代码和近似对象，均可在GitHub上获取，见\cite{code}。
英文摘要
We consider the defocusing quintic wave equation $\partial_t^2u-Δu+u^5=0$ for real-valued functions on $\mathbb{R}^{1+10}$, which is energy-supercritical. We construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-1/2}W^*\big(ω^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2π$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb{R}^{10}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first real-valued blow-up solutions of defocusing energy-supercritical NLW. Existence of the real profile is proved by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients, with rigorous error control. We find the approximate profile by numerically continuing rotating self-similar profiles of the complex-valued equation. This step is not essential for the blow-up proof, so it is left as non-rigorous numerics. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.
