Non-parabolic diffusion problems in one space dimension

TL;DR

This paper investigates non-parabolic diffusion problems in one dimension with complex flux behaviors, constructing weak solutions that reveal novel properties using advanced mathematical techniques like convex integration.

Contribution

It introduces a new approach to construct weak solutions for non-parabolic diffusion equations via inhomogeneous PDE inclusions, convex integration, and Baire category methods.

Findings

Solutions exhibit anomalous asymptotic behaviors

Energy dissipation and allocation properties are characterized

The method applies to various flux types like Perona-Malik and non-Fourier

Abstract

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.