Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

TL;DR

This paper establishes existence and uniqueness of very weak solutions for linear diffusion equations with singular absorption and unbounded flow, using weighted spaces and boundary distance techniques, with new results on boundary condition independence.

Contribution

It introduces a novel approach using weighted spaces based on boundary distance to prove solution uniqueness without boundary conditions under certain growth conditions.

Findings

Proved existence and uniqueness of very weak solutions.

Established boundary condition independence for certain potential growth.

Analyzed solution differentiability and gradient integrability issues.

Abstract

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.