Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy

Abstract

We are concerned with the identification of the diffusion coefficient u(x) in a strongly degenerate parabolic diffusion equation. The strong degeneracy means that ${u \in W^{1,\infty}}$, u vanishes at an interior point of the space domain and ${\frac{1}{u} \notin L^{1}.}$ The aim is to identify u from certain observations on the solution, by treating the identification problem as a nonlinear optimal control problem with the control in coefficients. The requirements related to the strong degeneracy of the equation impose to search the control u in ${W^{1, \infty}}$, restriction which represents a novelty and induces a particular difficulty in the determination of the optimality conditions. We prove the existence of a control and compute the optimality conditions both for homogeneous Dirichlet and Dirichlet–Neumann boundary conditions associated to the state system. In the case with a final time observation and homogeneous Dirichlet–Neumann boundary conditions, a very explicit form of the control and its uniqueness are provided by technical arguments.