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

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 $rac{1}{u} otin L^1$. 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.