标题： 一维空间中的非抛物扩散问题 显示英文标题

标题： Non-parabolic diffusion problems in one space dimension

摘要： 我们研究了一维空间中一些非抛物型扩散问题，其中扩散流表现出Perona-Malik、Höllig或非傅里叶类型的前向和后向性质。这类问题的经典弱解被构造出来以捕捉一些预期和意想不到的性质，包括异常渐近行为以及能量耗散或分配。解的具体性质将取决于扩散流的类型，但我们研究的主要方法依赖于将涉及的扩散方程重新表述为非齐次偏微分包含，并通过凸积分与Baire范畴法的结合来构建解。在此过程中，我们引入了偏微分包含的适当次解概念及其过渡规范，这在处理所构造弱解的一些特定特征时起着关键作用。 显示英文摘要

摘要： We study some non-parabolic diffusion problems in one-space dimension, where the diffusion flux exhibits forward and backward nature of the Perona-Malik, H\"ollig or non-Fourier type. Classical weak solutions to such problems are constructed in a way to capture some expected and unexpected properties, including anomalous asymptotic behaviors and energy dissipation or allocation. Specific properties of solutions will depend on the type of the diffusion flux, but the primary method of our study relies on reformulating diffusion equations involved as an inhomogeneous partial differential inclusion and on constructing solutions from the differential inclusion by a combination of the convex integration and Baire's category methods. In doing so, we introduce the appropriate notion of subsolutions of the partial differential inclusion and their transition gauge, which plays a pivotal role in dealing with some specific features of the constructed weak solutions.