AI 中文总结
该研究提出计算机辅助框架，将有限维尖点映射公式扩展至无限维，结合Newton-Kantorovich论证与谱估计，严格验证抛物型半线性偏微分方程的尖点分岔，并在两类方程的四个维度场景中完成证明。
AI 中文摘要
本文提出一种计算机辅助框架，用于严格验证抛物型半线性偏微分方程中空间对称平稳周期模式的尖点分岔。我们将此前在有限维中开发的尖点映射公式扩展至无限维情形，将尖点条件表述为傅里叶系数希尔伯特空间上的寻零问题，其非退化解对应尖点分岔点。关键技术环节是对所得序列空间中的伴随乘法算子进行精细处理，这比有限维情形更为复杂。从数值计算得到的近似解出发，我们构造性地运用Newton-Kantorovich论证，证明附近存在尖点映射的零点且具有局部唯一性。该解的非退化性直接导出三次范式系数c非零。为完成验证，我们严格包围线性化算子的谱，通过适配问题对称性结构的Gershgorin型估计，确认恰好有一个特征值具有零实部。进一步而言，尖点处严格正实部的特征值数目k与c的符号（均由该框架严格认证）共同决定尖点区域内三个共存解的稳定性结构：当k=0且c<0时出现双稳态，当k=0且c>0时出现单稳态，当k≥1时不存在稳定解。我们将该方法应用于Swift-Hohenberg方程和Gray-Scott系统，在一维和二维空间维度下均获得尖点分岔的严格证明。
英文摘要
In this paper, we present a computer-assisted framework for the rigorous validation of cusp bifurcations of spatially symmetric stationary periodic patterns in parabolic semilinear partial differential equations. Our approach extends to an infinite-dimensional setting the cusp map formulation previously developed in finite dimensions. We formulate the cusp conditions as a zero-finding problem on a Hilbert space of Fourier coefficients, whose non-degenerate solutions correspond to cusp bifurcation points. A key technical ingredient is a careful treatment of the adjoint multiplication operator in the resulting sequence space, which is more involved than in the finite-dimensional case. Starting from a numerically computed approximation, we develop a constructive Newton-Kantorovich argument to prove the existence and local uniqueness of a nearby zero of the cusp map. The non-degeneracy of this solution directly yields the non-vanishing of the cubic normal form coefficient $c$. To complete the verification, we rigorously enclose the spectrum of the linearized operator, confirming that exactly one eigenvalue has zero real part via Gershgorin-type estimates adapted to the symmetry structure of the problem. As a further consequence, the number $k$ of eigenvalues with strictly positive real part at the cusp and the sign of $c$ (both rigorously certified by the framework) together determine the stability structure of the three coexisting solutions inside the cusp region: bistability arises when $k=0$ and $c<0$, monostability when $k=0$ and $c>0$, and no stable solution exists when $k \geq 1$. We apply the method to the Swift--Hohenberg equation and the Gray--Scott system in both one and two spatial dimensions, obtaining rigorous proofs of cusp bifurcations in all four settings.
