Uniqueness for an ill-posed parabolic system
\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 concentration \(c(\cdot, \cdot)\). He reduces this to a question of injectivity of a certain map, which he proves.
scientific article; zbMATH DE number 5986968