A control approach for an identication problem associated to a strongly degenerate parabolic system with interior degeneracy

We study an identification problem associated with a strongly degenerate parabolic evolution equation of the type yt − Ay = f(t, x), (t, x) ∈ Q := (0, T) × (0, L) equipped with Dirichlet boundary conditions, where T > 0, L > 0, and f is in a suitable L2 space. The operator A has the form A1y =(uyx)x, or A2y = uyxx, and strong degeneracy means that the diffusion coefficient u ∈ W1,∞(0,L) satisfies u(x) > 0 except for an interior point of (0, L) and 1/u ∉ L1(0, L). Since an identification problem related to A1 was studied in Fragnelli et al. (J. Evol. Equ., 2014), here we devote more attention to the identification problem of u when A = A2. In this setting new weighted spaces of L2-type must be considered. Our techniques are based on the minimization problem of a functional depending on u, provided that some observations are known. Optimality conditions are also given.