Carleman estimate for a one-dimensional system of m coupled parabolic PDEs with BV diffusion coefficients

Hichem Ramoul

doi:10.1186/s13661-014-0195-2

Abstract

This paper is devoted to deriving global Carleman estimate for a one-dimensional linear coupled parabolic system of m equations with bounded variations (BV) diffusion coefficients. This kind of estimate is a generalization of the scalar result (Le Rousseau in J. Differ. Equ. 233:417-447, 2007). The key ingredient is to derive a global Carleman estimate for piecewise- diffusion coefficients based on the construction of a suitable weight function. The Carleman estimate in the case of BV diffusion coefficients is then obtained using the approach of BV diffusion coefficients by piecewise-constant coefficients. This Carleman estimate is used to show the observability inequality which yields the controllability result. MSC: 35K40, 26A45, 93B07.

Highlights

Introduction and notationsIn this paper we deal with one-dimensional m coupled parabolic equations with bounded variations (BV ) diffusion coefficients.Let = (, ) ⊂ R be a one-dimensional bounded domain, and we assume that T >
The main goal of this paper is to prove a global Carleman estimate for the operator ∂t + A with an interior observation region ω × (, T), where ω is a non-empty open subset of and such that kj are of class C on ω
The major novelty of our work is to prove a global Carleman estimate in the case of BV diffusion coefficients kj ( ≤ j ≤ m) for the operator ∂t + A

Summary

Introduction

The authors introduced a non-smooth weight function β, assuming that it satisfies the same transmission condition as the solution of a parabolic equation. The paper [ ] is devoted to proving the stability result using the Carleman estimate (with the observation of only one component) based on an adequate choice of weight function which is the same for each equation of a parabolic system. The major novelty of our work is to prove a global Carleman estimate (with m observations) in the case of BV diffusion coefficients kj ( ≤ j ≤ m) for the operator ∂t + A.

Objectives

Results

Conclusion