Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity

TL;DR

This paper develops a framework for establishing the existence and uniqueness of very weak solutions to the heat equation with anisotropic, singular, and time-dependent diffusivity tensors, addressing challenges where traditional weak solutions may not exist.

Contribution

It introduces a novel approach using very weak solutions to handle anisotropic and singular diffusivity in the heat equation, expanding the theoretical understanding of such problems.

Findings

Existence of very weak solutions under singular diffusivity

Uniqueness of solutions in the very weak sense

Framework applicable to anisotropic, time-dependent diffusivity

Abstract

We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.