Simultaneous reconstruction of the heat radiative coefficient and initial temperature for nonlinear parabolic equations

In this paper, the inverse problem of simultaneously reconstructing the radiation coefficient and the initial temperature of the parabolic equation is studied. Unlike the general parabolic equation, this paper not only has a nonlinear source term, but also the boundary conditions are nonlinear. Based on the optimal control framework, the original problem is first transformed into an optimization problem, and the necessary conditions for the minimum value of the control functional are proved.Being different from other ordinary optimization problems, the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters. Particularly, since the status of the two unknown coefficients in the cost functional are different, the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem