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

Jesus Ildefonso Díaz; David Gómez-Castro; Jean Michel Rakotoson; Roger Temam

doi:10.3934/dcds.2018023

Article Contents

2018, Volume 38, Issue 2: 509-546. Doi: 10.3934/dcds.2018023

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

1.
Instituto de Matematica Interdisciplinar & Dpto. de Matematica Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3,28040 Madrid, Spain
2.
Université de Poitiers, Laboratoire de Mathématiques et Applications -UMR CNRS 7348 -SP2MI, France, Bd Marie et Pierre Curie, Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
3.
Institute for Scientific Computation and Applied Mathematics, Indiana University, Bloomington, Indiana 47405, USA

* Corresponding author: Jean-Michel.Rakotoson@math.univ-poitiers.fr
* Corresponding author: Jean-Michel.Rakotoson@math.univ-poitiers.fr

Received: April 2017

Published: November 2017

The research of D. G´omez-Castro was supported by a FPU fellowship from the Spanish government. The research of J.I. D´ıaz and D. G´omez-Castro was partially supported by the project ref. MTM 2014-57113-P of the DGISPI (Spain). Roger Temam was partially supported by NSF grant DMS 1510249 and by the Research Fund of Indiana University

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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 $\text{IR}^N$. 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.

Keywords:

Linear diffusion equations,
singular absorption potential,
unbounded convective flow,
no boundary conditions,
dual problem,
local Kato inequality,
distance to the boundary weighted spaces.

Mathematics Subject Classification: 35J75, 35J15, 35J25, 35J67, 35J10, 76M23.

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