Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation.

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W(Ω), with m> N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs $(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast })$ of (P) converges as ε goes to 0 to a pair ( u, y) which realizes the minimum in (P), and y is the solution to the original state system. An alternative approach to the control problem is done by considering two controls related between them by a certain elliptic problem. This approach leads to the determination of simpler conditions of optimality, but under an additional restriction upon the initial data of the direct problem. [ABSTRACT FROM AUTHOR]