Coefficient identification in parabolic equations

This paper is devoted to the class of inverse problems for coefficient identification with an adjoint problem approach related to nonlinear parabolic partial differential equations. The unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients. First, we present a coefficient-to-data mapping. Then, based on maximum principle and the adjoint problem of the direct problem, the integral identities will be obtained. Using these identities, we can show that the coefficient-to-data mapping is continuous and strictly monotone. Furthermore, an approximate solution to the inverse problem is constructed, and the error is analyzed. Finally, the applicability of the method is demonstrated in numerical examples.