On the heat equation with nonlinearity and singular anisotropic potential on the boundary

TL;DR

This paper develops a well-posedness theory for the heat equation in a half-space with nonlinear and singular anisotropic boundary potentials, allowing for critical potentials with multiple singularities, and explores solution properties like symmetry and positivity.

Contribution

It introduces a new approach to handle critical boundary potentials with singularities without relying on Kato and Hardy inequalities, expanding the understanding of such PDEs.

Findings

Established well-posedness in weak L^p spaces for complex boundary potentials.

Proved key linear estimates for boundary operators in the Duhamel formulation.

Analyzed qualitative properties such as self-similarity, positivity, and symmetry of solutions.

Abstract

This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis $\overrightarrow{Ox_{n}}$.