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

Abstract

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 _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m(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 _ε ) 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.