Uniqueness for an ill-posed parabolic system
type: Item
instance of: scholarly article
title: Uniqueness for an ill-posed parabolic system
zbMATH Open document ID: 1237.35158
author: Faker Ben Belgacem
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2011-12-07T00:00:00Z
review text: \noindent The author studies a system that governs the interaction between organic pollutants and the oxygen available in stream waters. It is a two dimensional system of parabolic equations in one spatial variable, with \(b\) denoting the biochemical oxygen demand and \(c\) denoting the dissolved oxygen. Let \(I\) be the segment \((0, \xi)\) and \(T>0\) be a fixed real number. The system for the concentrations \((b, c)\) is as follows: \[ \begin{matrix} & \partial_t b - (d_*b^{\prime})^{\prime} + r_* b = 0 \;&\text{in} \;I \times(0,T), \\ &\partial_tc - (d c^{\prime})^{\prime} + rc - r_*b = 0,\;&\text{in} \;I \times(0,T), \\ &b(x, 0 ) = c(x, 0) = 0 \;&\text{in} \;I, \\&b(0,t) = c(0,t) = 0 \;&\text{in} \;(0,T), \\ &(c(\xi, t), dc^{\prime}(\xi,t)) = (\alpha(t), \beta(t)) \;&\text{in} \;(0,T). \end{matrix} \] \noindent The dispersion coefficients \((d, d_*)\) and the reaction parameters \((r, r_*)\) belong to \(L^{\infty}(I)\), are positive and bounded away from zero. No boundary conditions are given on \(b(\cdot, \cdot) \) at \(x = \xi\) while \(c(\cdot, \cdot)\) satisfies both Dirichlet and Neumann conditions. The system turns out to be ill-posed. \noindent The author rewrites the system as a vector-valued system \[ \partial_t(b, c) + A(b, c) = (f,g), \] \noindent for which uniqueness is connected to the behavior of the resolvent \(R(\lambda, A) = ( \lambda + A)^{-1}\). He shows that the norm of the resolvent is bounded independent of \(\lambda\) as a map on vector pairs of \(L^2(I)\) functions, and therefore by a result in \textit{A. Pazy} [Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag (1983; Zbl 0516.47023)], the vector valued problem has at most one solution. \noindent The author uses this result to consider the inverse problem of determining the location and intensity of a point source of pollution when measurements are uniquely made on the dissolved oxygen con…
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 5986968
zbMATH Keywords: ill-posed
zbMATH Keywords: parabolic system
zbMATH Keywords: resolvent
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: j.crma.2011.10.006
OpenAlex ID: W2001862435
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DOI: 10.1016/J.CRMA.2011.10.006
description: scientific article; zbMATH DE number 5986968