Regularization of Cauchy abstract problem for a coupled system for nonlinear elliptic equations

In this paper, we consider a Cauchy problem of coupled elliptic equations. We show that the problem is ill-posed in the sense of Hadamard and a regularization is in order. Under some weak a priori assumptions on the exact solution, we propose a new regularization method to stabilize the problem when the source term is a global or local Lipschitz function. Furthermore, we also obtain error estimates between the regularized solution and the sought solution.