AI 中文总结
本文基于谱Barron空间与椭圆型偏微分方程的关联，研究该框架下热方程的适定性、后向热方程的对数型稳定性估计，还扩展到时间分数阶热方程，刻画热核并推导相关性质，丰富了演化PDE理论。
AI 中文摘要
谱Barron空间是一类基于L¹的Fourier-Lebesgue范数刻画的函数空间，因能以可控复杂度的浅层神经网络表示函数，在逼近理论中受到广泛关注。同时，近期理论进展已明确建立了这类函数空间与椭圆型偏微分方程正则性理论之间深刻的内在联系。基于这一基础关联，本研究系统全面地探讨了在谱Barron空间框架下建立的热方程的适定性，在源项和传导系数的适当假设下，严格证明了解的存在性、唯一性和稳定性等适定性核心内容。我们还研究了一类典型的抛物型逆问题——后向热方程，推导得到了对数型条件稳定性估计，据我们所知，这是谱Barron空间框架下逆问题的首个稳定性估计。此外，我们将分析结果扩展到更复杂的时间分数阶热方程情形，这类方程描述反常扩散现象并引入非局部时间记忆效应，在该扩展框架下，我们刻画了相应的热核，推导了衰减估计和正则性性质，从而丰富了谱Barron空间框架下演化偏微分方程的理论体系。
英文摘要
Spectral Barron spaces, characterized by an \(L^1\)-based Fourier-Lebesgue norm, have earned significant attention in approximation theory due to their remarkable capacity to represent functions via shallow neural networks with controlled complexity. Meanwhile, recent theoretical advances have firmly established an intrinsic and profound connection between these function spaces and the regularity theory of elliptic partial differential equations. Building upon this foundational interplay, the present work undertakes a systematic and comprehensive investigation into the well-posedness of heat equations formulated within the spectral Barron spaces framework. Specifically, we rigorously establish the core aspects of well-posedness, including the existence, uniqueness, and stability of solutions, under suitable assumptions on the source terms and conductivity coefficients. We also investigate a typical parabolic inverse problem, namely the backward heat equation, for which we derive a logarithmic conditional stability estimate. To the best of our knowledge, this constitutes the first stability estimate for inverse problems within the spectral Barron space setting. Moreover, we extend our analytical results to address the more intricate setting of time-fractional heat equations, which govern anomalous diffusion phenomena and introduce nonlocal temporal memory effects. In this extended context, we provide a characterization of the corresponding heat kernels, deriving decay estimates, regularity properties, thereby enriching the theoretical landscape of evolutionary PDEs within the spectral Barron spaces setting.
