English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper we prove the existence and uniqueness of very weak solutions tolinear diffusion equations involving a singular absorption potential and/or anunbounded convective flow on a bounded open set of $\mathbb R^N$. In most ofthe paper we consider homogeneous Dirichlet boundary conditions but we provethat when the potential function grows faster than the distance to the boundaryto the power -2 then no boundary condition is required to get the uniqueness ofvery weak solutions. This result is new in the literature and must bedistinguished from other previous results in which such uniqueness of solutionswithout any boundary condition was proved for degenerate diffusion operators(which is not our case). Our approach, based on the treatment on some distanceto the boundary weighted spaces, uses a suitable regularity of the solution ofthe associated dual problem which is here established. We also consider thedelicate question of the differentiability of the very weak solution and provethat some suitable additional hypothesis on the data is required sinceotherwise the gradient of the solution may not be integrable on the domain.

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