An inverse problem for a system of nonlinear parabolic equations

Abstract

Abstract The inverse problem for a system of nonlinear parabolic equations is considered in the present paper. Namely, it is required to restore the initial condition by a given time-average value of the solution to the system of the nonlinear parabolic equations. An exact in the order error estimate of the optimal method for solving the inverse problem through the error estimate for the corresponding linear problem is obtained. A stable approximate solution to the unstable nonlinear problem under study is constructed by means of the projection regularization method which consists of using the representation of the approximate solution as a partial sum of the Fourier series. An exact in the order estimate for the error of the projection regularization method is obtained on one of the standard correctness classes. As a consequence, it is proved the optimality of the projection regularization method. As an example of a nonlinear system of parabolic equations, which has important practical applications, a spatially distributed model of blood coagulation is considered.