Recovery of thermal conductivity in two-dimensional media with nonlinear source by optimizations

Consider the heat conduction process with a temperature-depended source modeled by a nonlinear parabolic equation. We aim to identify the thermal conductivity from the extra measurement. By introducing the cost functional with regularization terms, the inverse problem is reformulated as an optimization problem. We prove the existence of the minimizers of the cost functional, with a rigorous analysis on the convergence property of the minimizing sequence. An iterative algorithm solving the optimization problem is proposed.