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

This paper deals with the determination of a pair (p,u)(p,u) in the nonlinear parabolic equation ut−uxx+p(x)f(u)=0,ut−uxx+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)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.