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

This paper deals with the determination of a pair (p, u) in the nonlinear parabolic equation u(t) - u(xx) + p(x)f(u) = 0, with initial and boundary conditions u(x, 0) = phi(x), u vertical bar(x=0) = u vertical bar(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. (C) 2009 Elsevier Ltd. All rights reserved.