Very weak solutions to the boundary-value problem of the homogenous heat equation

TL;DR

This paper develops a theory for very weak solutions to the homogeneous heat equation in bounded domains, focusing on low-regularity function spaces where traditional boundary data extensions are not feasible.

Contribution

It introduces a new framework for very weak solutions in $L_{p,q}$ spaces, providing existence and estimates without relying on boundary data extension or integration by parts.

Findings

Solutions exist in low-regularity Sobolev spaces

Key estimates are derived in the half-space setting

Existence results are obtained via perturbation and fixed point methods

Abstract

We consider the homogeneous heat equation in a domain $\Omega$ in $\mathbb{R}^n$ with vanishing initial data and the Dirichlet boundary condition. We are looking for solutions in $W^{r,s}_{p,q}(\Omega\times(0,T))$, where $r < 2$, $s < 1$, $1 \leq p < \infty$, $1 \leq q \leq \infty$. Since we work in the $L_{p,q}$ framework any extension of the boundary data and integration by parts are not possible. Therefore, the solution is represented in integral form and is referred as \emph{very weak} solution. The key estimates are performed in the half-space and are restricted to $L_q(0,T;W^{\alpha}_p(\Omega))$, $0 \leq \alpha < \frac{1}{p}$ and $L_q(0,T;W^{\alpha}_p(\Omega))$, $\alpha \leq 1$. Existence and estimates in the bounded domain $\Omega$ follow from a perturbation and a fixed point arguments.