Summary: We study an identification problem associated with a strongly degenerate parabolic evolution equation of the type \[ y_{t} -Ay = f(t,x), \quad (t,x) \in Q:= (0,T) \times (0,L) \] equipped with Dirichlet boundary conditions, where \(T > 0\), \(L > 0\), and \(f\) is in a suitable \(L^{2}\) space. The operator \(A\) has the form \(A_{1}y = (uy_{x})_{x}\), or \(A_{2}y = uy_{xx}\), and strong degeneracy means that the diffusion coefficient \(u \in W^{1, \infty}(0, L)\) satisfies \(u(x) > 0\) except for an interior point of \((0, L)\) and \(\frac{1} {u}\not\in L^{1}(0,L)\). Since an identification problem related to \(A_{1}\) was studied in Fragnelli et al. (J. Evol. Equ., 2014), here we devote more attention to the identification problem of \(u\) when \(A = A_{2}\). In this setting new weighted spaces of \(L^{2}\)-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.
For the entire collection see [Zbl 1357.35004].