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

TL;DR

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

Contribution

It introduces the concept of very weak solutions for the heat equation with singular, anisotropic, and time-dependent diffusivity, addressing the limitations of standard weak solutions.

Findings

Existence of very weak solutions under singular diffusivity conditions

Uniqueness of solutions in the very weak framework

Applicable to anisotropic and time-dependent diffusivity tensors

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.