An inverse problem of identifying the coefficient in a nonlinear parabolic equation

Abstract

This paper deals with the determination of a pair ( p , u ) in the nonlinear parabolic equation u t − u x x + p ( x ) f ( u ) = 0 , with initial and boundary conditions u ( x , 0 ) = ϕ ( x ) , u ∣ x = 0 = u ∣ x = 1 = 0 , from the overspecified data u ( x , T ) = g ( x ) . Based on the optimal control framework, the problem is transformed into a nonlinear optimization problem and the existence of the minimizer for the control functional is established. The necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Since the optimal control problem is nonconvex, one may not expect a unique solution in general. However, the local uniqueness and stability of the solution are proved, which is also the main contribution of the paper.